Generalized Convex Functions and Their Applications
aa r X i v : . [ m a t h . C A ] A p r Generalized Convex Functions and Their Applications
Adem Kili¸cman a, and Wedad Saleh ba Institute for Mathematical Research, University Putra Malaysia, Malaysiae-mail : [email protected] b Department of Mathematics, Putra University of Malaysia, Malaysiae-mail : wed 10 [email protected]
Abstract :
This study focuses on convex functions and their general-ized. Thus, we start this study by giving the definition of convex functionsand some of their properties and discussing a simple geometric property.Then we generalize E-convex functions and establish some their properties.Moreover, we give generalized s -convex functions in the second sense andpresent some new inequalities of generalized Hermite-Hadamard type forthe class of functions whose second local fractional derivatives of order α inabsolute value at certain powers are generalized s -convex functions in thesecond sense. At the end, some examples that these inequalities are ableto be applied to some special means are showed. Let M ⊆ R be an interval. A function ϕ : M ⊆ R −→ R is called aconvex if for any y , y ∈ M and η ∈ [0 , ϕ ( ηy + (1 − η ) y ) ≤ ηϕ ( y ) + (1 − η ) ϕ ( y ) . (1.1)If the inequality (1.1) is the strict inequality, then ϕ is called a strict convexfunction.From a geometrical point of view, a function ϕ is convex provided thatthe line segment connecting any two points of its graph lies on or abovethe graph. The function ϕ is strictly convex provided that the line segment Corresponding author email: [email protected] ( Adem Kili¸cman) onnecting any two points of its graph lies above the graph. If − ϕ is con-vex (resp. strictly convex), then ϕ is called concave (resp. strictly concave).The convexity of functions have been widely used in many branchesof mathematics, for example in mathematical analysis, function theory,functional analysis, optimization theory and so on. For aproduction func-tion x = ϕ ( L ), concacity of ϕ is expressed economically by saying that ϕ exhibits diminishing returns. While if ϕ is convex, then it exhibits in-creasing returns. Due to its applications and significant importance, theconcept of convexity has been extended and generalized in several direc-tions, see([2, 12, 23].Recently, the fractal theory has received significantly remarkable atten-tion from scientists and engineers. In the sense of Mandelbrot, a fractalset is the one whose Hausdorff dimension strictly exceeds the topologicaldimension[10, 19]. Many researchers studied the properties of functions onfractal space and constructed many kinds of fractional calculus by using dif-ferent approaches [2, 6, 31]. Particularly, in [27], Yang stated the analysisof local fractional functions on fractal space systematically, which includeslocal fractional calculus and the monotonicity of function.Throughout this chapter R α will be denoted a real linear fractal set. Definition 1.1. [15] A function ϕ : M ⊂ R −→ R α is called generalizedconvex if ϕ ( ηy + (1 − η ) y ) ≤ η α ϕ ( y ) + (1 − η ) α ϕ ( y ) (1.2)for all y , y ∈ M , η ∈ [0 ,
1] and α ∈ ( 0 ,
1] .It is called strictly generalized convex if the inequality (1.2) holds strictlywhenever y and y are distinct points and η ∈ (0 , − ϕ is generalizedconvex (respectively, strictly generalized convex), then ϕ is generalized con-cave (respectively, strictly generalized concave).In α = 1, we have a convex function ,i.e, (1.1) is obtained.Let f ∈ a I ( α ) a be a generalized convex function on [ a , a ] with a < a .Then, f (cid:18) a + a (cid:19) ≤ Γ(1 + α )( a − a ) α a I ( α ) a f ( x ) ≤ f ( a ) + f ( a )2 α . (1.3) s known as generalized Hermite-Hadmard’s inequality [14]. Many autherspaid attention to the study of generalized Hermite-Hadmard’s inequalityand generalized convex function, see [4, 16]. If α = 1 in (1.3), then [8] f (cid:18) a + a (cid:19) ≤ a − a Z a a f ( x ) dx ≤ f ( a ) + f ( a )2 , (1.4)which is known as classical Hermite-Hadamard inequality, for more prop-erties about this inequality we refer the interested readers to [9, 13]. In 1999, Youness [30] introduced E-convexity of sets and functions, whichhave some important applications in various branches of mathematicalsciences [1, 20]. However, Yang [26] showed that some results given byYouness [30] seem to be incorrect. Chen [7] extended E-convexity to asemi E-convexity and discussed some of its properties. For more results onE-convex function or semi E-convex function see [3, 11, 17, 18, 25].
Definition 2.1. [30](i) A set B ⊆ R n is called a E-convex iff there exists E : R n −→ R n suchthat ηE ( r ) + (1 − η ) E ( r ) ∈ B, ∀ r , r ∈ B, η ∈ [0 , . (ii) A function g : R n −→ R is called E-convex (ECF) on a set B ⊆ R n iff there exists E : R n −→ R n and g ( ηE ( r )+(1 − η ) E ( r ) ≤ ηg ( E ( r ))+(1 − η ) g ( E ( r )) , ∀ r , r ∈ B, η ∈ [0 , . The following propositions were proved in [30]:
Proposition 2.2. (i) Suppose that a set B ⊆ R n is E-convex, then E ( B ) ⊆ B .(ii) Assume that E ( B ) is convex and E ( B ) ⊆ B , then B is E-convex. Definition 2.3.
A function g : R n −→ R α is called a generalized E-convexfunction (gECF) on a set B ⊆ R n iff there exists a map E : R n −→ R n such that B is an E-convex set and g ( ηE ( r ) + (1 − η ) E ( r )) ≤ η α g ( E ( r )) + (1 − η ) α g ( E ( r )) , (2.1) r , r ∈ B, η ∈ (0 , and α ∈ ( 0 , On the other hand, if g ( ηE ( x ) + (1 − η ) E ( x )) ≥ η α g ( E ( x )) + (1 − η ) α g ( E ( x )) , ∀ x , x ∈ B, η ∈ (0 , and α ∈ ( 0 , , then g is called generalized E-concaveon B . If the inequality sings in the previous two inequality are strict, then g is called generalized strictly E-convex and generalized strictly E-concave,respectively. Proposition 2.4. (i) Every ECF on a convex set B is gECF , where E = I .(ii) If α = 1 in equation (2.1), then g is called ECF on a set B .(iii) If α = 1 and E = I in equation (2.1), then g is called a convexfunction The following two examples show that generalized E-convex functionwhich are not necessarily generalized convex.
Example 2.5.
Assume that B ⊆ R is given as B = (cid:8) ( x , x ) ∈ R : µ (0 ,
0) + µ (0 ,
3) + µ (2 , (cid:9) , with µ i > , X i =1 µ i = 1 and define a map E : R −→ R such as E ( x , x ) =(0 , x ) . The function g : R −→ R α defined by g ( x , x ) = ( x α ; x < ,x α x α ; x ≥ The function g is gECF on B, but is not generalized convex. Remark 2.6. If α −→ in the above example, then g goes to generalizedconvex function. Example 2.7.
Assume that g : R −→ R α is defined as g ( r ) = ( α ; r > , ( − r ) α ; r ≤ and assume that E : R −→ R is defined as E ( r ) = − r . Hence, R is anE-convex set and g is gECF, but is not generalized convex. heorem 2.1. Assume that B ⊆ R n is an E-convex set and g : B −→ R is an ECF. If g : U −→ R α is non-decreasing generalized convex functionsuch that the rang g ⊂ U , then g og is a gECF on B .Proof. Since g is ECF, then g ( ηE ( r ) + (1 − η ) E ( r )) ≤ ηg ( E ( r )) + (1 − η ) g ( E ( r )) , ∀ r , r ∈ B and η ∈ [0 , g is non-decreasing generalizedconvex function, then g og ( ηE ( r ) + (1 − η ) E ( r )) ≤ g [ ηg ( E ( r )) + (1 − η ) g ( E ( r ))] ≤ η α g ( g ( E ( r ))) + (1 − η ) α g ( g ( E ( r )))= η α g og ( E ( r )) + (1 − η ) α g og ( E ( r ))which implies that g og is a gECF on B .Similarly, g og is a strictly gECF if g is a strictly non-decreasinggeneralized convex function. Theorem 2.2.
Assume that B ⊆ R n is an E-convex set, and g i : B −→ R α , i = 1 , , ..., l are generalized E-convex function. Then, g = l X i =1 k αi g i is a generalized E-convex on B for all k αi ∈ R α Proof.
Since g i , i = 1 , , ..., l are gECF, then g i ( ηE ( r ) + (1 − η ) E ( r )) ≤ η α g i ( E ( r )) + (1 − η ) α g i ( E ( r )) , ∀ r , r ∈ B , η ∈ [0 ,
1] and α ∈ ( 0 ,
1] . Then, l X i =1 k αi g i ( ηE ( r ) + (1 − η ) E ( r )) ≤ η α l X i =1 k αi g i ( E ( r )) + (1 − η ) α l X i =1 k αi g i ( E ( r ))= η α g ( E ( r )) + (1 − η ) α g ( E ( r )) . Thus, g is a gECF. efinition 2.8. Assume that B ⊆ R n is a convex set. A function g : B −→ R α is called generalized quasi convex if g ( ηr + (1 − η ) r ) ≤ max { g ( r ) , g ( r ) } , ∀ r , r ∈ B and η ∈ [0 , . Definition 2.9.
Assume that B ⊆ R n is an E-convex set. A function g : B −→ R α is called(i) Generalized E-quasiconvex function iff g ( ηE ( r ) + (1 − η ) E ( r ) ≤ max { g ( E ( r )) , E ( g ( r )) } , ∀ r , r ∈ B and η ∈ [0 , .(ii) Strictly generalized E-quasiconcave function iff g ( ηE ( r ) + (1 − η ) E ( r ) > min { g ( E ( r )) , E ( g ( r )) } , ∀ r , r ∈ B and η ∈ [0 , . Theorem 2.3.
Assume that B ⊆ R n is an E-convex set, and g i : B −→ R α , i = 1 , , ..., l are gECF. Then,(i) The function g : B −→ R α which is defined by g ( r ) = sup i ∈ I g i ( r ) , r ∈ B is a gECF on B .(ii) If g i , i = 1 , , ..., l are generalized E-quasiconvex functions on B , thenthe function g is a generalized E-quasiconvex function on B .Proof. (i) Due to g i , i ∈ I be gECF on B , then g ( ηE ( r ) + (1 − η ) E ( r ))= sup i ∈ I g i ( ηE ( r ) + (1 − η ) E ( r )) ≤ η α sup i ∈ I g i ( E ( r )) + (1 − η ) α sup i ∈ I g i ( E ( r ))= η α g ( E ( r )) + (1 − η ) α g ( E ( r )) . Hence, g is a gECF on B . ii) Since g i , i ∈ I are generalized E-quasiconvex functions on B , then g ( ηE ( x ) + (1 − η ) E ( x )) = sup i ∈ I g i ( ηE ( x ) + (1 − η ) E ( x )) ≤ sup i ∈ I max { g i ( E ( x )) , g i ( E ( x )) } = max (cid:26) sup i ∈ I g i ( E ( x )) , sup i ∈ I g i ( E ( x )) (cid:27) = max { g ( E ( x )) , g ( E ( x )) } . Hence, g is a generalized E-quasiconvex function on B .Considering B ⊆ R n is a nonempty E-convex set. From Propostion2.2(i),we get E ( B ) ⊆ B .Hence, for any g : B −→ R α , the restriction ˜ g : E ( B ) −→ R α of g to E(B) defined by˜ g (˜ x ) = g (˜ x ) , ∀ ˜ x ∈ E ( B )is well defined. Theorem 2.4.
Assume that B ⊆ R n , and g : B −→ R α is a generalizedE-quasiconvex function on B . Then, the restriction ˜ g : U −→ R α of g toany nonempty convex subset U of E ( B ) is a generalized quasiconvex on U .Proof. Assume that x , x ∈ U ⊆ E ( B ), then there exist x ∗ , x ∗ ∈ B suchthat x = E ( x ∗ ) and x = E ( x ∗ ). Since U is a convex set, we have ηx + (1 − η ) x = ηE ( x ∗ ) + (1 − η ) E ( x ∗ ) ∈ U, ∀ η ∈ [0 , . Therefore, we have˜ g ( ηx + (1 − η ) x ) = ˜ g ( ηE ( x ∗ ) + (1 − η ) E ( x ∗ )) ≤ max { g ( E ( x ∗ )) , g ( E ( x ∗ )) } = max { g ( x ) , g ( x ) } = max { ˜ g ( x ) , ˜ g ( x ) } . Theorem 2.5.
Assume that B ⊆ R n is an E-convex set, and E ( B ) is aconvex set. Then, g : B −→ R α is a generalized E-quasiconvex on B iff itsrestriction ˜ g = g | E ( B ) is a generalized quasiconvex function on E ( B ) . roof. Due to Theorem2.4, the if condition is true. Conversely, supposethat x , x ∈ B , then E ( x ) , E ( x ) ∈ E ( B ) and ηE ( x ) + (1 − η ) E ( x ) ∈ E ( B ) ⊆ B, ∀ η ∈ [0 , E ( B ) ⊆ B , then g ( ηE ( x ) + (1 − η ) E ( x )) = ˜ g ( ηE ( x ) + (1 − η ) E ( x )) ≤ max { ˜ g ( E ( x )) , ˜ g ( E ( x )) } = max { g ( E ( x )) , g ( E ( x )) } . An analogous result to Theorem2.4 for the generalized E-convex case isas follows:
Theorem 2.6.
Assume that B ⊆ R n is an E-convex set, and g : B −→ R α is a gECF on B . Then, the restriction ˜ g : U −→ R α of g to any nonemptyconvex subset U of E ( B ) is a gCF. An analogous result to Theorem2.5 for the generalized E-convex case isas follows:
Theorem 2.7.
Assume that B ⊆ R n is an E-convex set, and E ( B ) is aconvex set. Then, g : B −→ R α is a gECF on B iff its restriction ˜ g = g | E ( B ) is a gCF on E ( B ) . The lower level set of goE : B −→ R α is defined as L r α ( goE ) = { x ∈ B : ( goE )( x ) = g ( E ( x )) ≤ r α , r α ∈ R α } . The lower level set of ˜ g : E ( B ) −→ R α is defined as L r α (˜ g ) = { ˜ x ∈ E ( B ) : ˜ g (˜ x ) = g (˜ x ) ≤ r α , r α ∈ R α } . Theorem 2.8.
Supose that E ( B ) be a convex set. A function g : B −→ R α is a generalized E-quasiconvex iff L r α (˜ g ) of its restriction ˜ g : E ( B ) −→ R α is a convex set for each r α ∈ R α .Proof. Due to E ( B ) be a convex set, then for each E ( x ) , E ( x ) ∈ E ( B ), wehave ηE ( x ) + (1 − η ) E ( x ) ∈ E ( B ) ⊆ B . Let ˜ x = E ( x ) and ˜ x = E ( x ). f ˜ x , ˜ x ∈ L r α (˜ g ), then g (˜ x ) ≤ r α and g (˜ x ) ≤ r α . Thus,˜ g ( η ˜ x + (1 − η )˜ x ) = g ( η ˜ x + (1 − η )˜ x )= g ( ηE ( x ) + (1 − η ) E ( x )) ≤ max { g ( E ( x )) , g ( E ( x )) } = max { g (˜ x ) , g (˜ x ) } = max { ˜ g (˜ x ) , ˜ g (˜ x ) }≤ r α . which show that η ˜ x + (1 − η )˜ x ∈ L r α (˜ g ). Hence, L r α (˜ g ) is a convex set.Conversely, let L r α (˜ g ) be a convex set for each r α ∈ R α , i.e., η ˜ x + (1 − η )˜ x ∈ L r α (˜ g ) , ∀ ˜ x , ˜ x ∈ L r α (˜ g ) and r α = max { g (˜ x ) , g (˜ x ) } . Thus, g ( ηE ( x ) + (1 − η ) E ( x )) = ˜ g ( ηE ( x ) + (1 − η ) E ( x ))= ˜ g ( η ˜ x + (1 − η )˜ x ) ≤ r α = max { g (˜ x ) , g (˜ x ) } = max { g ( E ( x )) , g ( E ( x )) } . Hence, g is a generalized E-quasiconvex. Theorem 2.9.
Let B ⊆ R n be a nonempty E-convex set and let g : B −→ R α be a generalized E-quasiconvex on B . Suppose that g : R α −→ R α is anon-decreasing function. Then, g og is a generalized E-quasiconvex.Proof. Since g : B −→ R α is generalized E-quasiconvex on B and g : R α −→ R α is a non-decreasing function, then( g og )( ηE ( x ) + (1 − η ) E ( x )) = g ( g ( ηE ( x ) + (1 − η ) E ( x ))) ≤ g (max { g ( E ( x )) , g ( E ( x )) } )= max { ( g og )( E ( x )) , ( g og )( E ( x )) } which shows that g og is a generalized E-quasiconvex on B . heorem 2.10. If the function g is a gECF on B ⊆ R n , then g is ageneralized E-quasiconvex on B .Proof. Assume that g is a gECF on B . Then, g ( ηE ( r ) + (1 − η ) E ( r )) ≤ η α g ( E ( r )) + (1 − η ) α g ( E ( r )) ≤ η α max { g ( E ( r )) , g ( E ( r )) } +(1 − η ) α max { g ( E ( r )) , g ( E ( r )) } = max { g ( E ( r )) , g ( E ( r )) } . E α -epigraph Definition 3.1.
Assume that B ⊆ R n × R α and E : R n −→ R n , then theset B is called E α -convex set iff ( ηE ( x ) + (1 − η ) E ( x ) , η α r α + (1 − η ) α r α ) ∈ B ∀ ( x , r α ) , ( x , r α ) ∈ B , η ∈ [0 , and α ∈ ( 0 , . Now, the E α - epigraph of g is given by epi E α ( g ) = { ( E ( x ) , r α ) : x ∈ B, r α ∈ R α , g ( E ( x )) ≤ r α } . A sufficient condition for g to be a gECF is given by the followingtheorem: Theorem 3.1.
Let E : R n −→ R n be an idempoted map. Assume that B ⊆ R n is an E-convex set and epi E α ( g ) is an E α -convex set where g : B −→ R α ,then g is a gECF on B .Proof. Assume that r , r ∈ B and ( E ( r ) , g ( E ( r ))) , ( E ( r ) , g ( E ( r ))) ∈ epi E α ( g ). Since epi E α ( g ) is E α - convex set, we have( ηE ( E ( r )) + (1 − η ) E ( E ( r )) , η α g ( E ( r )) + (1 − η ) α g ( E ( r ))) ∈ epi E α ( g ) , then g ( E ( ηE ( r )) + (1 − η ) E ( E ( r ))) ≤ η α g ( E ( r )) + (1 − η ) α g ( E ( r )) . Due to E : R n −→ R n be an idempotent map, then g ( ηE ( r ) + (1 − η ) E ( r )) ≤ η α g ( E ( r )) + (1 − η ) α g ( E ( r )) . Hence, g is a gECF. heorem 3.2. Assume that { B i } i ∈ I is a family of E α -convex sets. Then,their intersection ∩ i ∈ I B i is an E α -convex set.Proof. Considering ( x , r α ) , ( x , r α ) ∈ ∩ i ∈ I B i , then ( x , r α ) , ( x , r α ) ∈ B i , ∀ i ∈ I . By E α -convexity of B i , ∀ i ∈ I , then we have( ηE ( x ) + (1 − η ) E ( x ) , η α r α + (1 − η ) α r α ) ∈ B i , ∀ η ∈ [0 ,
1] and α ∈ ( 0 ,
1] . Hence,( ηE ( x ) + (1 − η ) E ( x ) , η α r α + (1 − η ) α r α ) ∈ ∩ i ∈ I B i . The following theorem is a special case of Theorem 2.3(i) where E : R n −→ R n is an idempotent map. Theorem 3.3.
Assume that E : R n −→ R n is an idempotent map, and B ⊆ R n is an E-convex set. Let { g i } i ∈ I be a family function which havebounded from above. If epi E α ( g i ) are E α -convex sets, then the function g which defined by g ( x ) = sup i ∈ I g i ( x ) , x ∈ B is a gECF on B .Proof. Since epi E α ( g i ) = { ( E ( x ) , r α ) : x ∈ B, r α ∈ R α , g i ( E ( x )) ≤ r α , i ∈ I } are E α - convex set in B × R α , then ∩ i ∈ I epi E α ( g i ) = { ( E ( x ) , r α ) : x ∈ B, r α ∈ R α , g i ( E ( x )) ≤ r α , i ∈ I } = { ( E ( x ) , r α ) : x ∈ B, r α ∈ R α , g ( E ( x )) ≤ r α } , (3.1)where g ( E ( x )) = sup i ∈ I g i ( E ( x )), also is E α -convex set. Hence, ∩ i ∈ I epi E α ( g i )is an E α -epigraph, then by Theorem3.2, g is a generalized E-convex func-tion on B . s -convex functions There are many researchers studied the properties of functions on fractalspace and constructed many kinds of fractional calculus by using differentapproaches see [5, 24, 28]In [14], two kinds of generlized s -convex functions on fractal sets areintroduced as follows: efinition 4.1. (i) A function ϕ : R + −→ R α , is called a generalized s -convex ( < s < )in the first sense if ϕ ( η y + η y ) ≤ η sα ϕ ( y ) + η sα ϕ ( y ) (4.1) for all y , y ∈ R + and all η , η ≥ with η s + η s = 1 , this class offunctions is denoted by GK s .(ii) A function ϕ : R + −→ R α , is called a generalized s -convex (
Assume that ϕ : R + −→ R + is a s-convex function in thesecond sense, < s < and y , y ∈ R + , y < y . If ϕ ∈ L ([ y , y ]) , then s − ϕ (cid:18) y + y (cid:19) ≤ y − y Z y y ϕ ( z ) dz ≤ ϕ ( y ) + ϕ ( y ) s + 1 . (4.2)If we set k = 1 s + 1 , then it is the best possible in the second inequalityin (4.2).A varation of generalized Hadamard’s inequality which holds for gen-eralized s -convex functions in the second sense [2]. Theorem 4.3.
Assume that ϕ : R + −→ R α + is a generalized s -convex func-tion in the second sense where < s < and y , y ∈ R + with y < y . If ϕ ∈ L ([ y , y ]) , then α ( s − ϕ (cid:18) y + y (cid:19) ≤ Γ(1 + α )( y − y ) α y I ( α ) y ϕ ( x ) ≤ Γ(1 + sα )Γ(1 + α )Γ(1 + ( s + 1) α ) ( ϕ ( y ) + ϕ ( y )) . (4.3) roof. We know that ϕ is generalized s -convex in the second sense,whichlead to ϕ ( ηy + (1 − η ) y ) ≤ η αs ϕ ( y ) + (1 − η ) αs ϕ ( y ) , ∀ η ∈ [0 , . Then the following inequality can be written:Γ(1 + α ) I ( α )1 ϕ ( ηy + (1 − η ) y ) ≤ ϕ ( y )Γ(1 + α ) I ( α )1 η αs + ϕ ( y )Γ(1 + α ) I ( α )1 (1 − η ) αs = Γ(1 + sα )Γ(1 + α )Γ(1 + ( s + 1) α ) ( ϕ ( y ) + y )By considering z = ηy + (1 − η ) y . ThenΓ(1 + α ) I ( α )1 ϕ ( ηy + (1 − η ) y ) = Γ(1 + α )( y − y ) α y I ( α ) y ϕ ( x )= Γ(1 + α )( y − y ) α y I ( α ) y ϕ ( z ) . Here Γ(1 + α )( y − y ) α y I ( α ) y ϕ ( x ) ≤ Γ(1 + sα )Γ(1 + α )Γ(1 + ( s + 1) α ) ( ϕ ( y ) + ϕ ( y )) . Then, the second inequality in (4.3) is given.Now ϕ (cid:18) z + z (cid:19) ≤ ϕ ( z ) + ϕ ( z )2 αs , ∀ z , z ∈ I. (4.4)Let z = ηy + (1 − η ) y and z = (1 − η ) y + ηy with η ∈ [0 , ϕ (cid:18) y + y (cid:19) ≤ ϕ ( ηy + (1 − η ) y ) + ϕ ((1 − η ) y + ηy )2 αs , ∀ η ∈ [0 , . So 1Γ(1 + α ) Z ϕ (cid:18) y + y (cid:19) ( dη ) α ≤ α ( s − ( y − y ) α y I ( α ) y ϕ ( z )Then, 2 α ( s − ϕ (cid:18) y + y (cid:19) ≤ Γ(1 + α )( y − y ) α y I ( α ) y ϕ ( z ) . emma 4.4. Assume that ϕ : [ y , y ] ⊂ R −→ R α is a local fractionalderivative of order α ( ϕ ∈ D α ) on ( y , y ) with y < y . If ϕ (2 α ) ∈ C α [ y , y ] , then the following equality holds: Γ(1 + 2 α )[Γ(1 + α )] α ( y − y ) y I ( α ) y ϕ ( x ) − Γ(1 + 2 α )2 α ϕ (cid:18) y + y (cid:19) = ( y − y ) α α (cid:20) I ( α )1 γ α ϕ (2 α ) (cid:18) γ y + y − γ ) y (cid:19) + I ( α )1 ( γ − α ϕ (2 α ) (cid:18) γy + (1 − γ ) y + y (cid:19)(cid:21) Proof.
From the local fractional integration by parts, we get B = 1Γ(1 + α ) Z γ α ϕ (2 α ) (cid:18) γ y + y − γ ) y (cid:19) ( dγ ) α = (cid:18) y − y (cid:19) α ϕ ( α ) (cid:18) y + y (cid:19) − Γ(1 + 2 α ) (cid:18) y − y (cid:19) α γ α ϕ (cid:18) γ y + y − γ ) y (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) +Γ(1 + 2 α )Γ(1 + α ) (cid:18) b − a (cid:19) α Z ϕ (cid:18) γ y + y − γ ) y (cid:19) ( dγ ) α = (cid:18) y − y (cid:19) α ϕ ( α ) (cid:18) y + y (cid:19) − Γ(1 + 2 α ) (cid:18) y − y (cid:19) α ϕ (cid:18) y + y (cid:19) +Γ(1 + 2 α )Γ(1 + α ) (cid:18) y − y (cid:19) α Z ϕ (cid:18) γ y + y − γ ) y (cid:19) ( dγ ) α Setting x = γ y + y − γ ) y , for γ ∈ [0 ,
1] and multiply the both sidesin the last equation by ( y − y ) α α , we get B = ( y − y ) α α I ( α )1 γ α ϕ (2 α ) (cid:18) γ y + y − γ ) y (cid:19) = ( y − y ) α α ϕ ( α ) (cid:18) y + y (cid:19) − Γ(1 + 2 α )4 α ϕ (cid:18) y + y (cid:19) + Γ(1 + 2 α )Γ(1 + α )2 α ( y − y ) α Z y y y ϕ ( x )( dx ) α . y the similar way, also we have B = ( y − y ) α α I ( α )1 ( γ − α ϕ (2 α ) (cid:18) γy + (1 − γ ) y + y (cid:19) = − ( y − y ) α α ϕ ( α ) (cid:18) y + y (cid:19) − Γ(1 + 2 α )4 α ϕ (cid:18) y + y (cid:19) + Γ(1 + 2 α )Γ(1 + α )2 α ( y − y ) α Z y y y ϕ ( x )( dx ) α . Thus, adding B and B , we get the desired result. Theorem 4.5.
Assume that ϕ : U ⊂ [ 0 , ∞ ) −→ R α such that ϕ ∈ D α on Int ( U ) ( Int ( U ) is the interior of U ) and ϕ (2 α ) ∈ C α [ y , y ] , where y , y ∈ U with y < y . If | ϕ | is generalized s -convex on [ y , y ] , for somefixed < s ≤ , then the following inequality holds: (cid:12)(cid:12)(cid:12)(cid:12) Γ(1 + 2 α )2 α ϕ (cid:18) y + y (cid:19) − Γ(1 + 2 α )[Γ(1 + α )] α ( y − y ) α y I ( α ) y ϕ ( x ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ ( y − y ) α α (cid:26) α Γ(1 + ( s + 2) α )Γ(1 + ( s + 3) α ) (cid:12)(cid:12)(cid:12)(cid:12) ϕ (2 α ) (cid:18) y + y (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) + (cid:20) Γ(1 + sα )Γ(1 + ( s + 1) α ) − α Γ(1 + ( s + 1) α )Γ(1 + ( s + 2) α ) + Γ(1 + ( s + 2) α )Γ(1 + ( s + 3) α ) (cid:21) h(cid:12)(cid:12)(cid:12) ϕ (2 α ) ( y ) (cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12) ϕ (2 α ) ( y ) (cid:12)(cid:12)(cid:12)i(cid:27) (4.5) ≤ ( y − y ) α α ( α (2 − s ) Γ(1 + ( s + 2) α )Γ(1 + ( s + 3) α ) Γ(1 + sα )Γ(1 + α )Γ(1 + ( s + 1) α ) + Γ(1 + sα )Γ(1 + ( s + 1) α ) − α Γ(1 + ( s + 1) α )Γ(1 + ( s + 2) α ) + Γ(1 + ( s + 2) α )Γ(1 + ( s + 3) α ) (cid:27) h(cid:12)(cid:12)(cid:12) ϕ (2 α ) ( y ) (cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12) ϕ (2 α ) ( y ) (cid:12)(cid:12)(cid:12)i . (4.6) roof. From Lemma 4.4, we have (cid:12)(cid:12)(cid:12)(cid:12)
Γ(1 + 2 α )2 α ϕ (cid:18) y + y (cid:19) − Γ(1 + 2 α )[Γ(1 + α )] α ( y − y ) α y I ( α ) y ϕ ( x ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ ( y − y ) α α (cid:20) I ( α )1 γ α (cid:12)(cid:12)(cid:12)(cid:12) ϕ (2 α ) ( γ y + y − γ ) y ) (cid:12)(cid:12)(cid:12)(cid:12) + I ( α )1 ( γ − α (cid:12)(cid:12)(cid:12)(cid:12) ϕ (2 α ) (cid:18) γy + (1 − γ ) y + y (cid:19)(cid:12)(cid:12)(cid:12)(cid:12)(cid:21) ≤ ( y − y ) α α I ( α )1 γ α (cid:20) γ αs (cid:12)(cid:12)(cid:12)(cid:12) ϕ (2 α ) (cid:18) y + y (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) + (1 − γ ) αs (cid:12)(cid:12)(cid:12) ϕ (2 α ) ( y ) (cid:12)(cid:12)(cid:12)(cid:21) + ( y − y ) α α I ( α )1 ( γ − α (cid:20) γ αs (cid:12)(cid:12)(cid:12) ϕ (2 α ) ( y ) (cid:12)(cid:12)(cid:12) + (1 − γ ) αs (cid:12)(cid:12)(cid:12)(cid:12) ϕ (2 α ) (cid:18) y + y (cid:19)(cid:12)(cid:12)(cid:12)(cid:12)(cid:21) = ( y − y ) α α (cid:26) Γ(1 + ( s + 2) α )Γ(1 + ( s + 3) α ) (cid:12)(cid:12)(cid:12)(cid:12) ϕ (2 α ) (cid:18) y + y (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) + (cid:20) Γ(1 + αs )Γ(1 + ( s + 1) α ) − α Γ(1 + ( s + 1) α )Γ(1 + ( s + 2) α ) + Γ(1 + ( s + 2) α )Γ(1 + ( s + 3) α ) (cid:21) (cid:12)(cid:12)(cid:12) ϕ (2 α ) ( y ) (cid:12)(cid:12)(cid:12)(cid:27) + ( y − y ) α α (cid:26) Γ(1 + ( s + 2) α )Γ(1 + ( s + 3) α ) (cid:12)(cid:12)(cid:12)(cid:12) ϕ (2 α ) (cid:18) y + y (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) + (cid:20) Γ(1 + αs )Γ(1 + ( s + 1) α ) − α Γ(1 + ( s + 1) α )Γ(1 + ( s + 2) α ) + Γ(1 + ( s + 2) α )Γ(1 + ( s + 3) α ) (cid:21) (cid:12)(cid:12)(cid:12) ϕ (2 α ) ( a ) (cid:12)(cid:12)(cid:12) + Γ(1 + ( s + 2) α )Γ(1 + ( s + 3) α ) (cid:12)(cid:12)(cid:12)(cid:12) ϕ (2 α ) (cid:18) y + y (cid:19)(cid:12)(cid:12)(cid:12)(cid:12)(cid:27) = ( y − y ) α α (cid:26) α Γ(1 + ( s + 2) α )Γ(1 + ( s + 3) α ) (cid:12)(cid:12)(cid:12)(cid:12) ϕ (2 α ) (cid:18) y + y (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) + (cid:20) Γ(1 + sα )Γ(1 + ( s + 1) α ) − α Γ(1 + ( s + 1) α )Γ(1 + ( s + 2) α ) + Γ(1 + ( s + 1) α )Γ(1 + ( s + 3) α ) (cid:21) h(cid:12)(cid:12)(cid:12) ϕ (2 α ) ( y ) (cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12) ϕ (2 α ) ( y ) (cid:12)(cid:12)(cid:12)i(cid:27) . This proves inequality (4.5). Since2 α ( s − ϕ (2 α ) (cid:18) y + y (cid:19) ≤ Γ(1 + sα )Γ(1 + α )Γ(1 + ( s + 1) α ) (cid:16) ϕ (2 α ) ( y ) + ϕ (2 α ) ( y ) (cid:17) , hen (cid:12)(cid:12)(cid:12)(cid:12) Γ(1 + 2 α )2 α ϕ (cid:18) y + y (cid:19) − Γ(1 + 2 α )[Γ(1 + α )] α ( y − y ) α y I ( α ) y ϕ ( x ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ ( y − y ) α α ( α Γ(1 + ( s + 2) α )Γ(1 + ( s + 3) α ) 2 − α ( s − Γ(1 + sα )Γ(1 + α )Γ(1 + ( s + 1) α ) h(cid:12)(cid:12)(cid:12) ϕ (2 α ) ( y ) (cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12) ϕ (2 α ) ( y ) (cid:12)(cid:12)(cid:12)i + (cid:20) Γ(1 + sα )Γ(1 + ( s + 1) α ) − α Γ(1 + ( s + 1) α )Γ(1 + ( s + 2) α ) + Γ(1 + ( s + 2) α )Γ(1 + ( s + 3) α ) (cid:21) h(cid:12)(cid:12)(cid:12) ϕ (2 α ) ( y ) (cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12) ϕ (2 α ) ( y ) (cid:12)(cid:12)(cid:12)i(cid:27) = ( y − y ) α α ( α (2 − s ) Γ(1 + ( s + 2) α )Γ(1 + ( s + 3) α ) Γ(1 + sα )Γ(1 + α )Γ(1 + ( s + 1) α ) + Γ(1 + sα )Γ(1 + ( s + 1) α ) − α Γ(1 + ( s + 1) α )Γ(1 + ( s + 2) α ) + Γ(1 + ( s + 2) α )Γ(1 + ( s + 3) α ) h(cid:12)(cid:12)(cid:12) ϕ (2 α ) ( y ) (cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12) ϕ (2 α ) ( y ) (cid:12)(cid:12)(cid:12)i(cid:27) Thus, we get the inequality (4.6)and the proof is complete.
Remark 4.6.
1. When α = 1 , Theorem 4.5 reduce to Theorem 2 in[21].2. If s = 1 in Theorem 4.5 , then (cid:12)(cid:12)(cid:12)(cid:12) Γ(1 + 2 α )2 α ϕ (cid:18) y + y (cid:19) − Γ(1 + 2 α )[Γ(1 + α )] α ( y − y ) α y I ( α ) y ϕ ( x ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ ( y − y ) α α (cid:26) α Γ(1 + 3 α )Γ(1 + 4 α ) (cid:12)(cid:12)(cid:12)(cid:12) ϕ (2 α ) (cid:18) y + y (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) + (cid:20) Γ(1 + α )Γ(1 + 2 α ) − α Γ(1 + 2 α )Γ(1 + 3 α ) + Γ(1 + 3 α )Γ(1 + 4 α ) (cid:21) h(cid:12)(cid:12)(cid:12) ϕ (2 α ) ( y ) (cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12) ϕ (2 α ) ( y ) (cid:12)(cid:12)(cid:12)i(cid:27) ≤ ( y − y ) α α ( α Γ(1 + 3 α )Γ(1 + 4 α ) [Γ(1 + α )] Γ(1 + 2 α ) + Γ(1 + α )Γ(1 + 2 α ) − α Γ(1 + 2 α )Γ(1 + 3 α ) + Γ(1 + 3 α )Γ(1 + 4 α ) (cid:27) h(cid:12)(cid:12)(cid:12) ϕ (2 α ) ( y ) (cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12) ϕ (2 α ) ( y ) (cid:12)(cid:12)(cid:12)i . (4.7)
3. If s = 1 and α = 1 in Theorem 4.5, then (cid:12)(cid:12)(cid:12)(cid:12) ϕ (cid:18) y + y (cid:19) − y − y Z y y ϕ ( x ) dx (cid:12)(cid:12)(cid:12)(cid:12) ≤ ( y − y ) (cid:26) (cid:12)(cid:12)(cid:12)(cid:12) ϕ ′′ (cid:18) y + y (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12) ϕ ′′ ( y ) (cid:12)(cid:12) + (cid:12)(cid:12) ϕ ′′ ( y ) (cid:12)(cid:12)(cid:27) ≤ ( y − y ) (cid:8)(cid:12)(cid:12) ϕ ′′ ( y ) (cid:12)(cid:12) + (cid:12)(cid:12) ϕ ′′ ( y ) (cid:12)(cid:12)(cid:9) e give a new upper bound of the left generalized Hadamard’s inequal-ity for generalized s -convex functions in the following theorem: Theorem 4.7.
Assume that ϕ : U ⊂ [ 0 , ∞ ) −→ R α such that ϕ ∈ D α on Int ( U ) and ϕ (2 α ) ∈ C α [ y , y ] , where y , y ∈ U with y < y . If | ϕ (2 α ) | p isgeneralized s -convex on [ y , y ] , for some fixed < s ≤ and p > with p + p = 1 , then the following inequality holds: (cid:12)(cid:12)(cid:12)(cid:12) Γ(1 + 2 α )2 α ϕ (cid:18) y + y (cid:19) − Γ(1 + 2 α )[Γ(1 + α )] α ( y − y ) α y I ( α ) y ϕ ( x ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ ( y − y ) α α (cid:20) Γ(1 + sα )Γ(1 + ( s + 1) α ) (cid:21) p (cid:20) Γ(1 + 2 p α )Γ(1 + (2 p + 1) α ) (cid:21) p × "(cid:18)(cid:12)(cid:12)(cid:12)(cid:12) ϕ (2 α ) (cid:18) y + y (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) p + (cid:12)(cid:12)(cid:12) ϕ (2 α ) ( y ) (cid:12)(cid:12)(cid:12) p (cid:19) p + (cid:18)(cid:12)(cid:12)(cid:12)(cid:12) ϕ (2 α ) (cid:18) y + y (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) p + (cid:12)(cid:12)(cid:12) ϕ (2 α ) ( y ) (cid:12)(cid:12)(cid:12) p (cid:19) p . (4.8) Proof.
Let p >
1, then from Lemma 4.4 and using generalized H¨older’sinequality [27], we obtain (cid:12)(cid:12)(cid:12)(cid:12)
Γ(1 + 2 α )2 α ϕ (cid:18) y + y (cid:19) − Γ(1 + 2 α )[Γ(1 + α )] α ( y − y ) α y I ( α ) y ϕ ( x ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ ( y − y ) α α (cid:26) I ( α )1 γ α (cid:12)(cid:12)(cid:12)(cid:12) ϕ (2 α ) (cid:18) γ y + y − γ ) y (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) + I ( α )1 ( γ − α (cid:12)(cid:12)(cid:12)(cid:12) ϕ (2 α ) (cid:18) γy + (1 − γ ) y + y (cid:19)(cid:12)(cid:12)(cid:12)(cid:12)(cid:27) ≤ ( y − y ) α α (cid:16) I ( α )1 γ p α (cid:17) p × (cid:18) I ( α )1 (cid:12)(cid:12)(cid:12)(cid:12) ϕ (2 α ) (cid:18) γ y + y − γ ) y (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) p (cid:19) p + ( y − y ) α α (cid:16) I ( α )1 (1 − γ ) p α (cid:17) p × (cid:18) I ( α )1 (cid:12)(cid:12)(cid:12)(cid:12) ϕ (2 α ) (cid:18) γy + (1 − γ ) y + y (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) p (cid:19) p . ince (cid:12)(cid:12) ϕ (2 α ) (cid:12)(cid:12) p is generalized s -convex, then I ( α )1 (cid:12)(cid:12)(cid:12)(cid:12) ϕ (2 α ) (cid:18) γ y + y − γ ) y (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) p ≤ Γ(1 + sα )Γ(1 + ( s + 1) α ) (cid:12)(cid:12)(cid:12)(cid:12) ϕ (2 α ) (cid:18) y + y (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) p + Γ(1 + sα )Γ(1 + ( s + 1) α ) (cid:12)(cid:12)(cid:12) ϕ (2 α ) ( y ) (cid:12)(cid:12)(cid:12) p , which means I ( α )1 (cid:12)(cid:12)(cid:12)(cid:12) ϕ (2 α ) (cid:18) γy + (1 − γ ) y + y (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) p ≤ Γ(1 + sα )Γ(1 + ( s + 1) α ) (cid:12)(cid:12)(cid:12) ϕ (2 α ) ( y ) (cid:12)(cid:12)(cid:12) p + Γ(1 + sα )Γ(1 + ( s + 1) α ) (cid:12)(cid:12)(cid:12)(cid:12) ϕ (2 α ) (cid:18) y + y (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) p . Hence (cid:12)(cid:12)(cid:12)(cid:12)
Γ(1 + 2 α )2 α ϕ (cid:18) y + y (cid:19) − Γ(1 + 2 α )[Γ(1 + α )] α ( y − y ) α y I ( α ) y ϕ ( x ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ ( y − y ) α α (cid:20) Γ(1 + sα )Γ(1 + ( s + 1) α ) (cid:21) p (cid:20) Γ(1 + 2 p α )Γ(1 + (2 p + 1) α ) (cid:21) p × ((cid:20)(cid:12)(cid:12)(cid:12)(cid:12) ϕ (2 α ) (cid:18) y + y (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) p + (cid:12)(cid:12)(cid:12) ϕ (2 α ) ( y ) (cid:12)(cid:12)(cid:12) p (cid:21) p + (cid:20)(cid:12)(cid:12)(cid:12)(cid:12) ϕ (2 α ) (cid:18) y + y (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) p + (cid:12)(cid:12)(cid:12) ϕ (2 α ) ( y ) (cid:12)(cid:12)(cid:12) p (cid:21) p ) . The proof is complete.
Remark 4.8. If s = 1 in Theorem 4.7, then (cid:12)(cid:12)(cid:12)(cid:12) Γ(1 + 2 α )2 α ϕ (cid:18) y + y (cid:19) − Γ(1 + 2 α )[Γ(1 + α )] α ( y − y ) α y I ( α ) y ϕ ( x ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ ( y − y ) α α (cid:20) Γ(1 + α )Γ(1 + 2 α ) (cid:21) p (cid:20) Γ(1 + 2 p α )Γ(1 + (2 p + 1) α ) (cid:21) p × ((cid:20)(cid:12)(cid:12)(cid:12)(cid:12) ϕ (2 α ) (cid:18) y + y (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) p + (cid:12)(cid:12)(cid:12) ϕ (2 α ) ( y ) (cid:12)(cid:12)(cid:12) p (cid:21) p + (cid:20)(cid:12)(cid:12)(cid:12)(cid:12) ϕ (2 α ) (cid:18) y + y (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) p + (cid:12)(cid:12)(cid:12) ϕ (2 α ) ( y ) (cid:12)(cid:12)(cid:12) p (cid:21) p ) . (4.9) orollary 4.9. Assume that ϕ : U ⊂ [ 0 , ∞ ) −→ R α such that ϕ ∈ D α on Int ( U ) and ϕ (2 α ) ∈ C α [ y , y ] , where y , y ∈ U with y < y . If | ϕ (2 α ) | p isgeneralized s -convex on [ y , y ] , for some fixed < s ≤ and p > with p + p = 1 , then the following inequality holds: (cid:12)(cid:12)(cid:12)(cid:12) Γ(1 + 2 α )2 α ϕ (cid:18) y + y (cid:19) − Γ(1 + 2 α )[Γ(1 + α )] α ( y − y ) α y I ( α ) y ϕ ( x ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ ( y − y ) α α [Γ(1 + sα )] p [Γ(1 + ( s + 1) α )] p (cid:20) Γ(1 + 2 p α )Γ(1 + (2 p + 1) α ) (cid:21) p × (cid:26)(cid:20)(cid:16) α (1 − s ) Γ(1 + sα )Γ(1 + α ) + Γ(1 + ( s + 1) α ) (cid:17) p +2 α (1 − s ) p [Γ(1 + α )] p [Γ(1 + α )] p (cid:21) h(cid:12)(cid:12)(cid:12) ϕ (2 α ) ( y ) (cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12) ϕ (2 α ) ( y ) (cid:12)(cid:12)(cid:12)i(cid:27) Proof.
Since (cid:12)(cid:12) ϕ (2 α ) (cid:12)(cid:12) p is generalized s -convex, then2 α ( s − ϕ (2 α ) (cid:18) y + y (cid:19) ≤ Γ(1 + sα )Γ(1 + α )Γ(1 + ( s + 1) α ) (cid:16) ϕ (2 α ) ( y ) + ϕ (2 α ) ( y ) (cid:17) . Hence using (4.8), we get (cid:12)(cid:12)(cid:12)(cid:12)
Γ(1 + 2 α )2 α ϕ (cid:18) y + y (cid:19) − Γ(1 + 2 α )[Γ(1 + α )] α ( y − y ) α y I ( α ) y ϕ ( x ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ ( y − y ) α α [Γ(1 + sα )] p [Γ(1 + ( s + 1) α )] p (cid:20) Γ(1 + 2 p α )Γ(1 + (2 p + 1) α ) (cid:21) p × nh(cid:16) α (1 − s ) Γ(1 + sα )Γ(1 + α ) + Γ(1 + ( s + 1) α ) (cid:17) | ϕ (2 α ) ( y ) | p +2 α (1 − s ) Γ(1 + sα )Γ(1 + α ) | ϕ (2 α ) ( y ) | p i p + h α (1 − s ) Γ(1 + sα )Γ(1 + α ) | ϕ (2 α ) ( y ) | p + (cid:16) α (1 − s ) Γ(1 + sα )Γ(1 + α ) + Γ(1 + ( s + 1) α ) (cid:17) | ϕ (2 α ) ( y ) | p i q (cid:27) and since k X i =1 ( x i + z i ) αn ≤ k X i =1 x αni + k X i =1 z αni or 0 < n < , x i , z i ≥ ∀ ≤ i ≤ k , then we have (cid:12)(cid:12)(cid:12)(cid:12) Γ(1 + 2 α )2 α ϕ (cid:18) y + y (cid:19) − Γ(1 + 2 α )[Γ(1 + α )] α ( y − y ) α y I ( α ) y ϕ ( x ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ ( y − y ) α α [Γ(1 + sα )] p [Γ(1 + ( s + 1) α )] p (cid:20) Γ(1 + 2 p α )Γ(1 + (2 p + 1) α ) (cid:21) p × (cid:26)(cid:20)(cid:16) α (1 − s ) Γ(1 + sα )Γ(1 + α ) + Γ(1 + ( s + 1) α ) (cid:17) p (cid:12)(cid:12)(cid:12) ϕ (2 α ) ( y ) (cid:12)(cid:12)(cid:12) +2 α (1 − s ) p [Γ(1 + sα )] p [Γ(1 + α )] p (cid:12)(cid:12)(cid:12) ϕ (2 α ) ( y ) (cid:12)(cid:12)(cid:12)(cid:21) + (cid:20) α (1 − s ) p [Γ(1 + sα )] p [Γ(1 + α )] p (cid:12)(cid:12)(cid:12) ϕ (2 α ) ( y ) (cid:12)(cid:12)(cid:12) + (cid:16) α (1 − s ) Γ(1 + sα )Γ(1 + α ) + Γ(1 + ( s + 1) α ) (cid:17) p (cid:12)(cid:12)(cid:12) ϕ (2 α ) ( y ) (cid:12)(cid:12)(cid:12)(cid:21)(cid:27) where 0 < p < p >
1. By a simple calculation, we obtain therequired result.Now, the generalized Hadamard’s type inequality for generalized s -concave functions. Theorem 4.10.
Assume that ϕ : U ⊂ [ 0 , ∞ ) −→ R α such that ϕ ∈ D α on Int ( U ) and ϕ (2 α ) ∈ C α [ y , y ] , where y , y ∈ U with y < y . If | ϕ (2 α ) | p isgeneralized s -convex on [ y , y ] , for some fixed < s ≤ and p > with p + p = 1 , then the following inequality holds: (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Γ(1 + 2 α )2 α ϕ (cid:18) y + y (cid:19) − Γ(1 + 2 α ) [Γ(1 + α )] α ( y − y ) α y I ( α ) y ϕ ( x ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ α ( s − p ( y − y ) α α (Γ(1 + α )) p (cid:20) Γ(1 + 2 p α )Γ(1 + (2 p + 1) α ) (cid:21) p × (cid:20)(cid:12)(cid:12)(cid:12)(cid:12) ϕ (2 α ) (cid:18) y + y (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12)(cid:12) ϕ (2 α ) (cid:18) y + 3 y (cid:19)(cid:12)(cid:12)(cid:12)(cid:12)(cid:21) roof. From Lemma 4.4 and using the generalized H¨older inequality for p > p + p = 1, we get (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Γ(1 + 2 α )2 α ϕ (cid:18) y + y (cid:19) − Γ(1 + 2 α ) [Γ(1 + α )] α ( a − a ) α y I ( α ) y ϕ ( x ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ ( y − y ) α α (cid:20) I ( α )1 γ α (cid:12)(cid:12)(cid:12)(cid:12) ϕ (2 α ) (cid:18) γ y + y − γ ) y (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) + I ( α )1 ( γ − α (cid:12)(cid:12)(cid:12)(cid:12) ϕ (2 α ) (cid:18) γy + (1 − γ ) y + y (cid:19)(cid:12)(cid:12)(cid:12)(cid:12)(cid:21) ≤ ( y − y ) α α (cid:16) I ( α )1 γ p α (cid:17) p (cid:18) I ( α )1 (cid:12)(cid:12)(cid:12)(cid:12) ϕ (2 α ) (cid:18) γ y + y − γ ) y (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) p (cid:19) p + ( y − y ) α α (cid:16) I ( α )1 ( γ − p α (cid:17) p (cid:18) I ( α )1 (cid:12)(cid:12)(cid:12)(cid:12) ϕ (2 α ) (cid:18) γy + (1 − γ ) y + y (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) p (cid:19) p . Since (cid:12)(cid:12) ϕ (2 α ) (cid:12)(cid:12) p is generalized s -concave, then I ( α )1 (cid:12)(cid:12)(cid:12)(cid:12) ϕ (2 α ) (cid:18) γ y + y − γ ) y (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) p ≤ α ( s − Γ(1 + α ) (cid:12)(cid:12)(cid:12)(cid:12) ϕ (2 α ) (cid:18) y + y (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) p (4.10)also I ( α )1 (cid:12)(cid:12)(cid:12)(cid:12) ϕ (2 α ) (cid:18) γy + (1 − γ ) y + y (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) p ≤ α ( s − Γ(1 + α ) (cid:12)(cid:12)(cid:12)(cid:12) ϕ (2 α ) (cid:18) y + 3 y (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) p . (4.11)From (4.10) and (4.11), we observe that (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Γ(1 + 2 α )2 α ϕ (cid:18) y + y (cid:19) − Γ(1 + 2 α ) [Γ(1 + α )] α ( y − y ) α y I ( α ) y ϕ ( x ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ ( y − y ) α α (cid:20) Γ(1 + 2 p α )Γ(1 + (2 p + 1) α ) (cid:21) p α ( s − p (Γ(1 + α )) p (cid:12)(cid:12)(cid:12)(cid:12) ϕ (2 α ) (cid:18) y + y (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) + ( y − y ) α α (cid:20) Γ(1 + 2 p α )Γ(1 + (2 p + 1) α ) (cid:21) p α ( s − p (Γ(1 + α )) p (cid:12)(cid:12)(cid:12)(cid:12) ϕ (2 α ) (cid:18) y + 3 y (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) = 2 α ( s − p ( y − y ) α α (Γ(1 + α )) p (cid:20) Γ(1 + 2 p α )Γ(1 + (2 p + 1) α ) (cid:21) p (cid:20)(cid:12)(cid:12)(cid:12)(cid:12) ϕ (2 α ) (cid:18) y + y (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12)(cid:12) ϕ (2 α ) (cid:18) y + 3 y (cid:19)(cid:12)(cid:12)(cid:12)(cid:12)(cid:21) the proof is complete. emark 4.11.
1. If α = 1 in Theorem 4.10, then (cid:12)(cid:12)(cid:12)(cid:12) ϕ (cid:18) y + y (cid:19) − y − y Z y y ϕ ( x ) dx (cid:12)(cid:12)(cid:12)(cid:12) ≤ s − q ( y − y ) (cid:20) p + 1) (cid:21) p (cid:20)(cid:12)(cid:12)(cid:12)(cid:12) ϕ ′′ (cid:18) y + y (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12)(cid:12) ϕ ′′ (cid:18) y + 3 y (cid:19)(cid:12)(cid:12)(cid:12)(cid:12)(cid:21)
2. If s = 1 and < h Γ(1+2 p α )Γ(1+(2 p +1) α ) i p < , p > in Theorem 4.10,then (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Γ(1 + 2 α )2 α ϕ (cid:18) y + y (cid:19) − Γ(1 + 2 α ) [Γ(1 + α )] α ( y − y ) α y I ( α ) y ϕ ( x ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ ( y − y ) α α (Γ(1 + α )) p (cid:20)(cid:12)(cid:12)(cid:12)(cid:12) ϕ (2 α ) (cid:18) y + y (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12)(cid:12) ϕ (2 α ) (cid:18) y + 3 y (cid:19)(cid:12)(cid:12)(cid:12)(cid:12)(cid:21) As in [22], some generalized means are considered such as : A ( y , y ) = y α + y α α , y , y ≥ L n ( y , y ) = h Γ(1+ nα )Γ(1+( n +1) α ) (cid:16) y ( n +1) α − y ( n +1) α (cid:17)i n , n ∈ Z {− , } , y , y ∈ R , y = y .In [14], the following example was given:let 0 < s < y α , y α , y α ∈ R α . Defining for x ∈ R + , ϕ ( b ) = ( y α b = 0 ,y α b sα + y α b > . If y α ≥ α and 0 α ≤ y α ≤ y α , then ϕ ∈ GK s . Proposition 5.1.
Let < y < y and s ∈ (0 , . Then (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Γ(1 + 2 α )2 α A s ( y , y ) − Γ(1 + 2 α ) [Γ(1 + α )] α ( y − y ) α L ss ( y , y ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ ( y − y ) α α (cid:12)(cid:12)(cid:12)(cid:12) Γ(1 + sα )Γ(1 + ( s − α ) (cid:12)(cid:12)(cid:12)(cid:12) ( α Γ(1 + 3 α ) [Γ(1 + α )] Γ(1 + 4 α )Γ(1 + 2 α )+ Γ(1 + α )Γ(1 + 2 α ) − α Γ(1 + 2 α )Γ(1 + 3 α ) + Γ(1 + 3 α )Γ(1 + 4 α ) (cid:27) h | y | ( s − α + | y | ( s − α i roof. The result follows from Remark 4.7 (2) with ϕ : [0 , −→ [0 α , α ], ϕ ( x ) = x sα . and when α = 1, we have the following inequalitly: (cid:12)(cid:12)(cid:12)(cid:12) A s ( y , y ) − y − y L ss ( y , y ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ ( y − y ) | s ( s − | (cid:8) | y | s − + | y | s − (cid:9) . (5.1) Proposition 5.2.
Let < y < y and s ∈ (0 , . Then (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Γ(1 + 2 α )2 α A s ( y , y ) − Γ(1 + 2 α ) [Γ(1 + α )] α ( y − y ) α L ss ( y , y ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ ( y − y ) α α (cid:20) Γ(1 + α )Γ(1 + 2 α ) (cid:21) p (cid:12)(cid:12)(cid:12)(cid:12) Γ(1 + sα )Γ(1 + ( s − α ) (cid:12)(cid:12)(cid:12)(cid:12) (cid:20) Γ(1 + 2 p α )Γ(1 + (2 p + 1) α ) (cid:21) p × (cid:12)(cid:12)(cid:12)(cid:12) y + y (cid:12)(cid:12)(cid:12)(cid:12) ( s − p α + | y | ( s − p α ! p + (cid:12)(cid:12)(cid:12)(cid:12) y + y (cid:12)(cid:12)(cid:12)(cid:12) ( s − p α + | y | ( s − p α ! p , where p > and p + p = 1 .Proof. The result follows (4.9) with ϕ : [0 , −→ [0 α , α ], ϕ ( x ) = x sα , andwhen α = 1, we have the following inequalitly: (cid:12)(cid:12)(cid:12)(cid:12) A s ( y , y ) − y − y L ss ( y , y ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ ( y − y ) | s ( s − | p p + 1) p (cid:12)(cid:12)(cid:12)(cid:12) y + y (cid:12)(cid:12)(cid:12)(cid:12) ( s − p + | y | ( s − p ! p + (cid:12)(cid:12)(cid:12)(cid:12) y + y (cid:12)(cid:12)(cid:12)(cid:12) ( s − p + | y | ( s − p ! p . (5.2)Where A ( y , y ) and L n ( y , y ) in (5.1) and (5.2) are known as1. Arithmetic mean: A ( y , y ) = y + y , y , y ∈ R + ; . Logarithmic mean : L ( y , y ) = y − y ln | y | − ln | y | , | y | 6 = y , y , a = 0 , y , y ∈ R + ;Generalized Log-mean: L n ( y , y ) = (cid:20) y n +12 − y n +11 ( n + 1)( y − y ) (cid:21) n , n ∈ Z \ {− , } , y , y ∈ R + . Now, we give application to wave equation on Cantor sets:the wave equation on Cantor sets (local fractional wave equation) was givenby [27] ∂ α f ( x, t ) ∂t α = A α ∂ α f ( x, t ) ∂x α (5.3)Following (5.3), a wave equation on Cantor sets was proposed as follows[29]: ∂ α f ( x, t ) ∂t α = x α Γ(1 + 2 α ) ∂ α f ( x, t ) ∂x α , ≤ α ≤ f ( x, t ) is a fractal wave function and the initial value is given by f ( x,
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