Inequalities via s-convexity and log-convexity
aa r X i v : . [ m a t h . C A ] S e p INEQUALITIES VIA s − CONVEXITY AND log − CONVEXITY
AHMET OCAK AKDEMIR ⋆ , MERVE AVCI ARDIC¸ ♦ , ♠ , AND M. EMIN ¨OZDEMIR H Abstract.
In this paper, we obtain some new inequalities for functions whosesecond derivatives’ absolute value is s − convex and log − convex. Also, we givesome applications for numerical integration. INTRODUCTION
We start with the well-known definition of convex functions: a function f : I → R , ∅ 6 = I ⊂ R , is said to be convex on I if inequality f ( tx + (1 − t ) y ) ≤ tf ( x ) + (1 − t ) f ( y )holds for all x, y ∈ I and t ∈ [0 , . In the paper [12], authors gave the class of functions which are s − convex in thesecond sense by the following way. A function f : [0 , ∞ ) → R is said to be s − convexin the second sence if f ( tx + (1 − t ) y ) ≤ t s f ( x ) + (1 − t ) s f ( y )holds for all x, y ∈ [0 , ∞ ) , t ∈ [0 ,
1] and for some fixed s ∈ (0 , . The class of s − convex functions in the second sense is usually denoted with K s . Besides in [12], Hudzik and Maligranda proved that if s ∈ (0 , f ∈ K s implies f ([0 , ∞ )) ⊆ [0 , ∞ ) , i.e., they proved that all functions from K s , s ∈ (0 , , arenonnegative. Example 1. ( [12] ) Let s ∈ (0 , and a, b, c ∈ R . We define function f : [0 , ∞ ) → R as f ( t ) = (cid:26) a, t = 0 ,bt s + c, t > . It can be easily checked that(i) If b ≥ and ≤ c ≤ a, then f ∈ K s , (ii) If b > and c < , then f / ∈ K s . Several researchers studied on s − convex functions, some of them can be foundin [12]-[17].Another kind of convexity is log − convexity that is mentioned in [6] by Niculescuas following. Mathematics Subject Classification.
Key words and phrases.
Convex function, s − convex function, log − convex function, Ostrowskiinequality, H¨older inequality, power-mean inequality.This study was supported by A˘grı ˙Ibrahim C¸ e¸cen University BAP with project numberFEF.14.011. ♠ Corresponding Author. ⋆ , MERVE AVCI ARDIC¸ ♦ , ♠ , AND M. EMIN ¨OZDEMIR H A positive function f is called log − convex on a real interval I = [ a, b ], if for all x, y ∈ [ a, b ] and λ ∈ [0 , f ( λx + (1 − λ ) y ) ≤ f λ ( x ) f − λ ( y ) . For recent results for log − convex functions, we refer to readers [2]-[9].Now, we give a motivated inequality for convex functions:Let f : I ⊂ R → R be a convex function on the interval I of real numbers and a, b ∈ I with a < b . The inequality1 b − a Z ba f ( x ) dx ≤ (cid:20) f (cid:18) a + b (cid:19) + f ( a ) + f ( b )2 (cid:21) is known as Bullen’s inequality for convex functions [8], p. 39.We also consider the following useful inequality:Let f : I ⊂ [0 , ∞ ] → R be a differentiable mapping on I ◦ , the interior of theinterval I , such that f ′ ∈ L [ a, b ] where a, b ∈ I with a < b . If | f ′ ( x ) | ≤ M , thenthe following inequality holds (see [11]).(1.1) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) f ( x ) − b − a Z ba f ( u ) du (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ Mb − a " ( x − a ) + ( b − x ) This inequality is well known in the literature as the Ostrowski inequality . The main aim of this paper is to prove some new integral inequalities for s − convexand log − convex functions by using the integral identity that is obtained by Sarıkayaand Set in [1]. We also give some applications to our results in numerical integra-tion. Some of our results are similar to the Ostrowski inequality and for specialselections of the parameters, we proved some new inequalities of Bullen’s type.2. inequalities for s − convex functions We need the following Lemma which is obtained by Sarıkaya and Set in [1], soas to prove our results:
Lemma 1.
Let f : [ a, b ] → R be an absolutely continuous mapping. Denote by K ( x, . ) : [ a, b ] → R the kernel given by K ( x, t ) = αα + β ( t − a )( x − t ) x − a , t ∈ [ a, x ] − βα + β ( b − t )( x − t ) b − x , t ∈ [ x, b ] where α, β ∈ R nonnegative and not both zero, then the identity Z ba K ( x, t ) f ′′ ( t ) dt = f ( x ) + αf ( a ) + βf ( b ) α + β − α + β " αx − a Z xa f ( t ) dt + βb − x Z bx f ( t ) dt holds. Theorem 1.
Let f : [ a, b ] → R be an absolutely continuous mapping such that f ′′ ∈ L [ a, b ] . If | f ′′ | is s − convex in the second sense on [ a, b ] for some fixed s ∈ (0 , , NEQUALITIES VIA s − CONVEXITY AND log − CONVEXITY 3 then (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) f ( x ) + αf ( a ) + βf ( b ) α + β − α + β " αx − a Z xa f ( t ) dt + βb − x Z bx f ( t ) dt ≤ αα + β ( x − a ) ( s + 2) ( s + 3) [ | f ′′ ( x ) | + | f ′′ ( a ) | ]+ βα + β ( b − x ) ( s + 2) ( s + 3) [ | f ′′ ( x ) | + | f ′′ ( b ) | ] holds where α, β ∈ R nonnegative and not both zero.Proof. From Lemma 1, using the property of the modulus and s − convexity of | f ′′ | , we can write (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) f ( x ) + αf ( a ) + βf ( b ) α + β − α + β " αx − a Z xa f ( t ) dt + βb − x Z bx f ( t ) dt ≤ Z ba | K ( x, t ) | | f ′′ ( t ) | dt ≤ Z xa αα + β x − a | t − a | | x − t | | f ′′ ( t ) | dt + Z bx βα + β b − x | b − t | | x − t | | f ′′ ( t ) | dt = α ( α + β ) ( x − a ) Z xa ( t − a ) ( x − t ) (cid:12)(cid:12)(cid:12)(cid:12) f ′′ (cid:18) t − ax − a x + x − tx − a a (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) dt + β ( α + β ) ( b − x ) Z bx ( b − t ) ( t − x ) (cid:12)(cid:12)(cid:12)(cid:12) f ′′ (cid:18) t − xb − x b + b − tb − x x (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) dt ≤ α ( α + β ) ( x − a ) Z xa ( t − a ) ( x − t ) (cid:20)(cid:18) t − ax − a (cid:19) s | f ′′ ( x ) | + (cid:18) x − tx − a (cid:19) s | f ′′ ( a ) | (cid:21) dt + β ( α + β ) ( b − x ) Z bx ( b − t ) ( t − x ) (cid:20)(cid:18) t − xb − x (cid:19) s | f ′′ ( b ) | + (cid:18) b − tb − x (cid:19) s | f ′′ ( x ) | (cid:21) dt = αα + β ( x − a ) ( s + 2) ( s + 3) [ | f ′′ ( x ) | + | f ′′ ( a ) | ] + βα + β ( b − x ) ( s + 2) ( s + 3) [ | f ′′ ( x ) | + | f ′′ ( b ) | ]where we use the fact that Z xa ( t − a ) s +1 ( x − t ) dt = Z xa ( t − a ) ( x − t ) s +1 dt = ( x − a ) s +3 ( s + 2) ( s + 3)and Z bx ( b − t ) ( t − x ) s +1 dt = Z bx ( b − t ) s +1 ( t − x ) dt = ( b − x ) s +3 ( s + 2) ( s + 3) . The proof is completed. (cid:3)
AHMET OCAK AKDEMIR ⋆ , MERVE AVCI ARDIC¸ ♦ , ♠ , AND M. EMIN ¨OZDEMIR H Corollary 1.
Suppose that all the assumptions of Theorem 1 are satisfied with | f ′′ | ≤ M. Then we have (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) f ( x ) + αf ( a ) + βf ( b ) α + β − α + β " αx − a Z xa f ( t ) dt + βb − x Z bx f ( t ) dt ≤ M ( s + 2) ( s + 3) " α ( x − a ) + β ( b − x ) α + β . Corollary 2.
In Theorem 1, if we choose α = β = 1 , we obtain (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) f ( x ) + f ( a ) + f ( b )2 − " x − a Z xa f ( t ) dt + 1 b − x Z bx f ( t ) dt ≤ ( x − a ) + ( b − x ) s + 2) ( s + 3) | f ′′ ( x ) | + 12 ( s + 2) ( s + 3) h ( x − a ) | f ′′ ( a ) | + ( b − x ) | f ′′ ( b ) | i . Corollary 3.
In Theorem 1, if we choose α = β = and x = a + b , we obtain thefollowing Bullen type inequality; (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:20) f (cid:18) a + b (cid:19) + f ( a ) + f ( b )2 (cid:21) − b − a Z ba f ( t ) dt (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ ( b − a ) s + 2) ( s + 3) (cid:20)(cid:12)(cid:12)(cid:12)(cid:12) f ′′ (cid:18) a + b (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) + | f ′′ ( a ) | + | f ′′ ( b ) | (cid:21) . Theorem 2.
Let f : [ a, b ] → R be an absolutely continuous mapping such that f ′′ ∈ L [ a, b ] . If | f ′′ | q is s − convex in the second sense on [ a, b ] for some fixed s ∈ (0 , and q > with p + q = 1 , then (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) f ( x ) + αf ( a ) + βf ( b ) α + β − α + β " αx − a Z xa f ( t ) dt + βb − x Z bx f ( t ) dt ≤ (cid:18) αα + β (cid:19) p ( x − a ) q ( s + 1) q ( β ( p + 1 , p + 1)) p (cid:2) | f ′′ ( x ) | q + | f ′′ ( a ) | q (cid:3) q + (cid:18) βα + β (cid:19) p ( b − x ) q ( s + 1) q ( β ( p + 1 , p + 1)) p (cid:2) | f ′′ ( b ) | q + | f ′′ ( x ) | q (cid:3) q where β ( x, y ) = R t x − (1 − t ) y − dt, x, y > is the Euler Beta function, α, β ∈ R nonnegative and not both zero. NEQUALITIES VIA s − CONVEXITY AND log − CONVEXITY 5
Proof.
From Lemma 1, using the property of the modulus, H¨older inequality and s − convexity of | f ′′ | q , we can write (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) f ( x ) + αf ( a ) + βf ( b ) α + β − α + β " αx − a Z xa f ( t ) dt + βb − x Z bx f ( t ) dt ≤ (cid:18)Z xa (cid:18) αα + β ( t − a ) ( x − t ) x − a (cid:19) p dt (cid:19) p (cid:18)Z xa (cid:12)(cid:12)(cid:12)(cid:12) f ′′ (cid:18) t − ax − a x + x − tx − a a (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) q dt (cid:19) q + Z bx (cid:18) βα + β ( b − t ) ( x − t ) b − x (cid:19) p dt ! p Z bx (cid:12)(cid:12)(cid:12)(cid:12) f ′′ (cid:18) t − xb − x b + b − tb − x x (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) q dt ! q ≤ αα + β ( x − a ) (cid:18)Z xa ( t − a ) p ( x − t ) p ( x − a ) p ( x − a ) p dt (cid:19) p (cid:18)Z xa (cid:12)(cid:12)(cid:12)(cid:12) f ′′ (cid:18) t − ax − a x + x − tx − a a (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) q dt (cid:19) q + βα + β ( b − x ) Z bx ( b − t ) p ( x − t ) p ( b − x ) p ( b − x ) p dt ! p Z bx (cid:12)(cid:12)(cid:12)(cid:12) f ′′ (cid:18) t − xb − x b + b − tb − x x (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) q dt ! q ≤ αα + β ( x − a ) ( β ( p + 1 , p + 1)) p (cid:20)Z xa (cid:18) t − ax − a (cid:19) s | f ′′ ( x ) | q + (cid:18) x − tx − a (cid:19) s | f ′′ ( a ) | q (cid:21) q + βα + β ( b − x ) ( β ( p + 1 , p + 1)) p "Z bx (cid:18) t − xb − x (cid:19) s | f ′′ ( b ) | q + (cid:18) b − tb − x (cid:19) s | f ′′ ( x ) | q q . We get the desired result by making use of the necessary computation. (cid:3)
Theorem 3.
Under the assumptions of Theorem 2, the following inequality (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) f ( x ) + αf ( a ) + βf ( b ) α + β − α + β " αx − a Z xa f ( t ) dt + βb − x Z bx f ( t ) dt ≤ β ( p + 1 , p + 1) p (cid:18)(cid:18) αα + β (cid:19) p ( x − a ) p + (cid:18) βα + β (cid:19) p ( b − x ) p (cid:19) p × (cid:18) b − as + 1 (cid:19) q (cid:0) | f ′′ ( a ) | q + | f ′′ ( b ) | q (cid:1) q holds where β ( x, y ) is the Euler Beta function. AHMET OCAK AKDEMIR ⋆ , MERVE AVCI ARDIC¸ ♦ , ♠ , AND M. EMIN ¨OZDEMIR H Proof.
From Lemma 1, using the property of the modulus, H¨older inequality and s − convexity of | f ′′ | q , we can write (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) f ( x ) + αf ( a ) + βf ( b ) α + β − α + β " αx − a Z xa f ( t ) dt + βb − x Z bx f ( t ) dt ≤ Z ba | K ( x, t ) | p dt ! p Z ba | f ′′ ( t ) | q dt ! q = Z xa (cid:18) αα + β ( t − a ) ( x − t )( x − a ) ( x − a ) ( x − a ) (cid:19) p dt + Z bx (cid:18) βα + β ( b − t ) ( x − t )( b − x ) ( b − x ) ( b − x ) (cid:19) p dt ! p × Z ba (cid:12)(cid:12)(cid:12)(cid:12) f ′′ (cid:18) t − ab − a b + b − tb − a a (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) q dt ! q . We get the desired result by making use of the necessary computation. (cid:3)
The next result is obtained by using the well-known power-mean integral in-eqaulity:
Theorem 4.
Let f : [ a, b ] → R be an absolutely continuous mapping such that f ′′ ∈ L [ a, b ] . If | f ′′ | q is s − convex in the second sense on [ a, b ] for some fixed s ∈ (0 , and q ≥ , then (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) f ( x ) + αf ( a ) + βf ( b ) α + β − α + β " αx − a Z xa f ( t ) dt + βb − x Z bx f ( t ) dt ≤ αα + β ( x − a ) − q [( s + 2) ( s + 3)] q (cid:2) | f ′′ ( x ) | q + | f ′′ ( a ) | q (cid:3) q + βα + β ( b − x ) − q [( s + 2) ( s + 3)] q (cid:2) | f ′′ ( b ) | q + | f ′′ ( x ) | q (cid:3) q holds where α, β ∈ R nonnegative and not both zero. NEQUALITIES VIA s − CONVEXITY AND log − CONVEXITY 7
Proof.
From Lemma 1, using the property of the modulus, power-mean integralinequality and s − convexity of | f ′′ | q , we can write (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) f ( x ) + αf ( a ) + βf ( b ) α + β − α + β " αx − a Z xa f ( t ) dt + βb − x Z bx f ( t ) dt ≤ α ( α + β ) ( x − a ) (cid:18)Z xa ( t − a ) ( x − t ) dt (cid:19) − q × (cid:18)Z xa ( t − a ) ( x − t ) (cid:18)(cid:18) t − ax − a (cid:19) s | f ′′ ( x ) | q + (cid:18) x − tx − a (cid:19) s | f ′′ ( a ) | q (cid:19) dt (cid:19) q + β ( α + β ) ( b − x ) Z bx ( b − t ) ( t − x ) dt ! − q × Z bx ( t − b ) ( t − x ) (cid:18)(cid:18) t − xb − x (cid:19) s | f ′′ ( b ) | q + (cid:18) b − tb − x (cid:19) s | f ′′ ( x ) | q (cid:19) dt ! q = α ( α + β ) ( x − a ) ( x − a ) ! − q ( x − a ) ( s + 2) ( s + 3) (cid:0) | f ′′ ( x ) | q + | f ′′ ( a ) | q (cid:1)! q + β ( α + β ) ( b − x ) ( b − x ) ! − q ( b − x ) ( s + 2) ( s + 3) (cid:0) | f ′′ ( b ) | q + | f ′′ ( x ) | q (cid:1)! q . The proof is completed. (cid:3)
Remark 1.
In Theorem 4, if we choose q = 1 Theorem 4 reduces to Theorem 1.
Remark 2.
If we choose s = 1 for all the results, we obtain new results for convexfunctions. inequalities for log − convex functions In this section, we will give some results for log − convex functions. For thesimplicity, we will use the following notations: κ = (cid:18) | f ′′ ( x ) || f ′′ ( a ) | (cid:19) x − a τ = (cid:18) | f ′′ ( b ) || f ′′ ( x ) | (cid:19) b − x . Theorem 5.
Let f : [ a, b ] → R be an absolutely continuous mapping such that f ′′ ∈ L [ a, b ] . If | f ′′ | is log − convex function on [ a, b ] and κ = 1 , τ = 1 , then (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) f ( x ) + αf ( a ) + βf ( b ) α + β − α + β " αx − a Z xa f ( t ) dt + βb − x Z bx f ( t ) dt ≤ α ( α + β ) ( x − a ) (cid:18) | f ′′ ( a ) | x | f ′′ ( x ) | a (cid:19) x − a (cid:18) κ a − κ x ) − ( a − x ) ( κ a + κ x ) log κ log κ (cid:19) + β ( α + β ) ( b − x ) | f ′′ ( x ) | b | f ′′ ( b ) | x ! b − x τ x − τ b + ( b − x ) (cid:0) τ b + τ x (cid:1) log τ log τ ! AHMET OCAK AKDEMIR ⋆ , MERVE AVCI ARDIC¸ ♦ , ♠ , AND M. EMIN ¨OZDEMIR H holds where κ = 1 , τ = 1 , α, β ∈ R nonnegative and not both zero.Proof. From Lemma 1 and by using the log − convexity of | f ′′ | , we can write (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) f ( x ) + αf ( a ) + βf ( b ) α + β − α + β " αx − a Z xa f ( t ) dt + βb − x Z bx f ( t ) dt ≤ Z xa αα + β x − a | t − a | | x − t | | f ′′ ( t ) | dt + Z bx βα + β b − x | b − t | | x − t | | f ′′ ( t ) | dt = α ( α + β ) ( x − a ) Z xa ( t − a ) ( x − t ) (cid:12)(cid:12)(cid:12)(cid:12) f ′′ (cid:18) t − ax − a x + x − tx − a a (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) dt + β ( α + β ) ( b − x ) Z bx ( b − t ) ( t − x ) (cid:12)(cid:12)(cid:12)(cid:12) f ′′ (cid:18) t − xb − x b + b − tb − x x (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) dt ≤ α ( α + β ) ( x − a ) Z xa ( t − a ) ( x − t ) h | f ′′ ( x ) | t − ax − a | f ′′ ( a ) | x − tx − a i dt + β ( α + β ) ( b − x ) Z bx ( b − t ) ( t − x ) h | f ′′ ( b ) | t − xb − x | f ′′ ( x ) | b − tb − x i dt = α ( α + β ) ( x − a ) (cid:18) | f ′′ ( a ) | x | f ′′ ( x ) | a (cid:19) x − a Z xa ( t − a ) ( x − t ) κ t dt + β ( α + β ) ( b − x ) | f ′′ ( x ) | b | f ′′ ( b ) | x ! b − x Z bx ( b − t ) ( t − x ) τ t dt. By a simple computation, we get the result. (cid:3)
Corollary 4.
In Theorem 5, if we choose α = β = 1 , we obtain (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) f ( x ) + f ( a ) + f ( b )2 − " x − a Z xa f ( t ) dt + 1 b − x Z bx f ( t ) dt ≤
12 ( x − a ) (cid:18) | f ′′ ( a ) | x | f ′′ ( x ) | a (cid:19) x − a (cid:18) κ a − κ x ) − ( a − x ) ( κ a + κ x ) log κ log κ (cid:19) + 12 ( b − x ) | f ′′ ( x ) | b | f ′′ ( b ) | x ! b − x τ x − τ b + ( b − x ) (cid:0) τ b + τ x (cid:1) log τ log τ ! . NEQUALITIES VIA s − CONVEXITY AND log − CONVEXITY 9
Corollary 5.
In Theorem 5, if we choose α = β = and x = a + b , we obtain thefollowing Bullen type inequality; (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:20) f (cid:18) a + b (cid:19) + f ( a ) + f ( b )2 (cid:21) − b − a Z ba f ( t ) dt (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ | f ′′ ( a ) | a + b (cid:12)(cid:12) f ′′ (cid:0) a + b (cid:1)(cid:12)(cid:12) a ! b − a κ a − κ a + b + (cid:0) b − a (cid:1) (cid:16) κ a + κ a + b (cid:17) log κ ( b − a ) log κ + (cid:12)(cid:12) f ′′ (cid:0) a + b (cid:1)(cid:12)(cid:12) b | f ′′ ( b ) | a + b ! b − a τ a + b − τ b + (cid:0) b − a (cid:1) (cid:16) τ b + τ a + b (cid:17) log τ ( b − a ) log τ where κ = (cid:12)(cid:12) f ′′ (cid:0) a + b (cid:1)(cid:12)(cid:12) | f ′′ ( a ) | ! b − a τ = | f ′′ ( b ) | (cid:12)(cid:12) f ′′ (cid:0) a + b (cid:1)(cid:12)(cid:12) ! b − a . Theorem 6.
Let f : [ a, b ] → R be an absolutely continuous mapping such that f ′′ ∈ L [ a, b ] . If | f ′′ | q is log − convex function on [ a, b ] and q > with p + q = 1 , then (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) f ( x ) + αf ( a ) + βf ( b ) α + β − α + β " αx − a Z xa f ( t ) dt + βb − x Z bx f ( t ) dt ≤ αα + β ( x − a ) ( β ( p + 1 , p + 1)) p (cid:18) | f ′′ ( a ) | x | f ′′ ( x ) | a (cid:19) x − a κ qxx − a − κ qax − a log κ qx − a ! q + βα + β ( b − x ) ( β ( p + 1 , p + 1)) p | f ′′ ( x ) | b | f ′′ ( b ) | x ! b − x τ qbb − x − τ qxb − x log τ qb − x ! q . where β ( x, y ) = R t x − (1 − t ) y − dt, x, y > is the Euler Beta function and κ = 1 , τ = 1 , α, β ∈ R nonnegative and not both zero. ⋆ , MERVE AVCI ARDIC¸ ♦ , ♠ , AND M. EMIN ¨OZDEMIR H Proof.
From Lemma 1, by using log − convexity of | f ′′ | q and by applying H¨olderinequality , we get (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) f ( x ) + αf ( a ) + βf ( b ) α + β − α + β " αx − a Z xa f ( t ) dt + βb − x Z bx f ( t ) dt ≤ (cid:18)Z xa (cid:18) αα + β ( t − a ) ( x − t ) x − a (cid:19) p dt (cid:19) p (cid:18)Z xa (cid:12)(cid:12)(cid:12)(cid:12) f ′′ (cid:18) t − ax − a x + x − tx − a a (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) q dt (cid:19) q + Z bx (cid:18) βα + β ( b − t ) ( x − t ) b − x (cid:19) p dt ! p Z bx (cid:12)(cid:12)(cid:12)(cid:12) f ′′ (cid:18) t − xb − x b + b − tb − x x (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) q dt ! q ≤ αα + β ( x − a ) (cid:18)Z xa ( t − a ) p ( x − t ) p ( x − a ) p ( x − a ) p dt (cid:19) p (cid:18)Z xa (cid:12)(cid:12)(cid:12)(cid:12) f ′′ (cid:18) t − ax − a x + x − tx − a a (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) q dt (cid:19) q + βα + β ( b − x ) Z bx ( b − t ) p ( x − t ) p ( b − x ) p ( b − x ) p dt ! p Z bx (cid:12)(cid:12)(cid:12)(cid:12) f ′′ (cid:18) t − xb − x b + b − tb − x x (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) q dt ! q ≤ αα + β ( x − a ) ( β ( p + 1 , p + 1)) p (cid:18) | f ′′ ( a ) | x | f ′′ ( x ) | a (cid:19) qx − a Z xa | f ′′ ( x ) | qx − a | f ′′ ( a ) | qx − a ! t dt q + βα + β ( b − x ) ( β ( p + 1 , p + 1)) p | f ′′ ( x ) | b | f ′′ ( b ) | x ! qb − x Z bx | f ′′ ( b ) | qb − x | f ′′ ( x ) | qb − x ! t dt q . By computing the above integrals, we get the desired result. (cid:3)
Theorem 7.
Let f : [ a, b ] → R be an absolutely continuous mapping such that f ′′ ∈ L [ a, b ] . If | f ′′ | q is log − convex function on [ a, b ] and q ≥ , then (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) f ( x ) + αf ( a ) + βf ( b ) α + β − α + β " αx − a Z xa f ( t ) dt + βb − x Z bx f ( t ) dt ≤ α ( x − a ) − q − q ( α + β ) (cid:18) | f ′′ ( a ) | x | f ′′ ( x ) | a (cid:19) x − a (cid:18) κ qa − κ qx ) − q ( a − x ) ( κ qa + κ qx ) log κ log κ q (cid:19) q + β ( b − x ) − q − q ( α + β ) | f ′′ ( x ) | b | f ′′ ( b ) | x ! b − x τ qx − τ qb + q ( b − x ) (cid:0) τ qb + τ qx (cid:1) log τ log τ q ! q holds where κ q = 1 , τ q = 1 , α, β ∈ R nonnegative and not both zero. NEQUALITIES VIA s − CONVEXITY AND log − CONVEXITY 11
Proof.
From Lemma 1, by using the well-known power-mean integral inequalityand log − convexity of | f ′′ | q , we have (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) f ( x ) + αf ( a ) + βf ( b ) α + β − α + β " αx − a Z xa f ( t ) dt + βb − x Z bx f ( t ) dt ≤ α ( α + β ) ( x − a ) (cid:18)Z xa ( t − a ) ( x − t ) dt (cid:19) − q (cid:18)Z xa ( t − a ) ( x − t ) (cid:16) | f ′′ ( x ) | q t − ax − a | f ′′ ( a ) | q x − tx − a (cid:17) dt (cid:19) q + β ( α + β ) ( b − x ) Z bx ( b − t ) ( t − x ) dt ! − q Z bx ( t − b ) ( t − x ) (cid:16) | f ′′ ( b ) | q t − xb − x | f ′′ ( x ) | q b − tb − x (cid:17) dt ! q = α ( α + β ) ( x − a ) ( x − a ) ! − q (cid:18) | f ′′ ( a ) | x | f ′′ ( x ) | a (cid:19) x − a (cid:18) κ qa − κ qx ) − q ( a − x ) ( κ qa + κ qx ) log κ log κ q (cid:19) q + β ( α + β ) ( b − x ) ( b − x ) ! − q | f ′′ ( x ) | b | f ′′ ( b ) | x ! b − x τ qx − τ qb + q ( b − x ) (cid:0) τ qb + τ qx (cid:1) log τ log τ q ! q . Which completes the proof. (cid:3)
Remark 3.
In Theorem 7, if we choose q = 1 Theorem 7 reduces to Theorem 5.
Corollary 6.
For the particular selections of the parameters α, β and the variable x, one can obtain several new inequalities for log − convex functions, we omit thedetails. APPLICATIONS FOR NUMERICAL INTEGRATION
Suppose that d = { a = x < x < ... < x n = b } is a partition of the interval [ a, b ] ,h i = x i +1 − x i , for i = 0 , , , ..., n − Z ba f ( x ) dx = A MT ( d, f ) + R MT ( d, f ) , where A MT ( π, f ) = 14 n − X i =0 h i (cid:20) f ( x i ) + 2 f (cid:18) x i + x i +1 (cid:19) + f ( x i +1 ) (cid:21) Here, the term R MT ( d, f ) denotes the associated approximation error. (See [10]) Proposition 1.
Let f : [ a, b ] → R be an absolutely continuous mapping such that f ′′ ∈ L [ a, b ] . If | f ′′ | is log − convex function on [ a, b ] and κ = 1 , τ = 1 , then forthe partition d, following inequality holds | R MT ( d, f ) |≤ | f ′′ ( x i ) | xi + xi +12 (cid:12)(cid:12)(cid:12) f ′′ (cid:16) x i + x i +1 (cid:17)(cid:12)(cid:12)(cid:12) x i hi κ x i i − κ xi + xi +12 i + (cid:0) h i (cid:1) (cid:18) κ x i i + κ xi + xi +12 i (cid:19) log κ i h i log κ i + (cid:12)(cid:12)(cid:12) f ′′ (cid:16) x i + x i +1 (cid:17)(cid:12)(cid:12)(cid:12) x i +1 | f ′′ ( x i +1 ) | xi + xi +12 hi τ xi + xi +12 i − τ x i +1 i + (cid:0) h i (cid:1) (cid:18) τ x i +1 i + τ xi + xi +12 i (cid:19) log τ i h i log τ i . ⋆ , MERVE AVCI ARDIC¸ ♦ , ♠ , AND M. EMIN ¨OZDEMIR H where κ i = 1 , τ i = 1 and defined as κ i = (cid:12)(cid:12)(cid:12) f ′′ (cid:16) x i + x i +1 (cid:17)(cid:12)(cid:12)(cid:12) | f ′′ ( x i ) | hi τ i = | f ′′ ( x i +1 ) | (cid:12)(cid:12)(cid:12) f ′′ (cid:16) x i + x i +1 (cid:17)(cid:12)(cid:12)(cid:12) hi . Proof.
By applying Corollary 5 to the subintervals [ x i , x i +1 ] of d, ( i = 0 , , ..., n − (cid:3) Proposition 2.
Let f : [ a, b ] → R be an absolutely continuous mapping suchthat f ′′ ∈ L [ a, b ] . If | f ′′ | is s − convex in the second sense on [ a, b ] for some fixed s ∈ (0 , , then for partition d of [ a, b ] the following inequality holds: | R MT ( d, f ) |≤ h i s + 2) ( s + 3) (cid:20)(cid:12)(cid:12)(cid:12)(cid:12) f ′′ (cid:18) x i + x i +1 (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) + | f ′′ ( x i ) | + | f ′′ ( x i +1 ) | (cid:21) . Proof.
By applying Corollary 3 to the subintervals [ x i , x i +1 ] of d, ( i = 0 , , ..., n − (cid:3) References [1] M. Z. Sarikaya and E. Set, On new Ostrowski type integral inequalities, Thai Journal ofMathematics, Vol. 12 (2014), No: 1, 145-154.[2] A.M. Fink, Hadamard’s inequality for log − concave functions, Math. Comput. Modelling32(5–6) (2000), 625–629.[3] B.G. Pachpatte, A note on integral inequalities involving two log − convex functions, Mathe-matical Inequalities & Applications, 7(4) (2004), 511-515.[4] B.G. Pachpatte, A note on Hadamard type integral inequalities involving several log − convexfunctions, Tamkang Journal of Mathematics, 36(1) (2005), 43-47.[5] C.E.M. Pearce and J. Peˇcari´c. Inequalities for differentiable mappings with application tospecial means and quadrature formulae, Appl. Math. Lett., 13(2) 2000, 51-55.[6] C.P. Niculescu, The Hermite–Hadamard inequality for log − convex functions, Nonlinear Anal-ysis, 75 (2012), 662–669.[7] G-S. Yang, K-L. Tseng and H-t. Wang, A note on integral inequalities of Hadamard type forlog − convex and log − concave functions, Taiwanese Journal of Mathematics, 16(2) (2012),479-496.[8] S.S. Dragomir & C. Pearce, Selected topics on Hermite-Hadamard inequali-ties and applications, Victoria University: RGMIA Monographs, (17) 2000.[http://ajmaa.org/RGMIA/monographs/hermite hadamard.html].[9] S.S. Dragomir, Some Jensen’s Type Inequalities for log − Convex Functions of SelfadjointOperators in Hilbert Spaces, Bulletin of the Malaysian Mathematical Sciences Society, 34(3)(2011), 445-454.[10] S.S. Dragomir, P. Cerone and J. Roumeliotis, A new generalization of Ostrowski’s integralinequality for mappings whose derivatives are bounded and applications in numerical inte-gration and for special means, Applied Mathematics Letters, 13 (2000), 19-25.[11] A. Ostrowski, ¨Uber die Absolutabweichung einer differentierbaren Funktion von ihren Inte-gralmittelwert, Comment. Math. Helv., 10, 226-227, (1938).[12] H. Hudzik, L. Maligranda, Some remarks on s − convex functions, Aequationes Math. 48(1994) 100-111.[13] S.S. Dragomir, S. Fitzpatrick, The Hadamard’s inequality for s − convex functions in thesecond sense. Demonstratio Math. 32 (4) (1999) 687-696. NEQUALITIES VIA s − CONVEXITY AND log − CONVEXITY 13 [14] U.S. Kırmacı, M.K. Bakula, M.E. ¨Ozdemir and J. Peˇcari´c, Hadamard-type inequalities for s − convex functions, Appl. Math. Comp., 193 (2007), 26-35.[15] S. Hussain, M.I. Bhatti and M. Iqbal, Hadamard-type inequalities for s − convex functions,Punjab University, Journal of Mathematics, 41 (2009) 51-60.[16] M. Avci, H. Kavurmaci and M.E. ¨Ozdemir, New inequalities of Hermite–Hadamard typevia s − convex functions in the second sense with applications, Appl. Math. and Comput.,217(2011) 5171-5176.[17] M.Z. Sarikaya, E. Set and M.E. ¨Ozdemir, On new inequalities of Simpson’s type for s − convexfunctions, Comp. and Math. with Appl., 60 (2010) 2191-2199. ⋆ A˘grı ˙Ibrahim C¸ ec¸en University, Faculty of Science and Arts, Department of Math-ematics, A˘grı, TURKEY
E-mail address : [email protected] ♦ Adıyaman University, Faculty of Science and Arts, Department of Mathematics,Adıyaman, TURKEY