On some inequalities for m- and (alpha,m)-logarithmically convex functions
aa r X i v : . [ m a t h . C A ] N ov ON SOME INEQUALITIES FOR m - AND ( α, m ) -LOGARITHMICALLY CONVEX FUNCTIONS MEVL ¨UT TUNC¸
Abstract.
In this paper, we establish some new integral inequalities for for m - and ( α, m )-logarithmically convex functions. Introduction
In [1], the concepts of m - and ( α, m )-logarithmically convex functions were in-troduced as follows. Definition 1.
A function f : [0 , b ] → (0 , ∞ ) is said to be m -logarithmically convexif the inequality (1.1) f ( tx + m (1 − t ) y ) ≤ [ f ( x )] t [ f ( y )] m (1 − t ) holds for all x, y ∈ [0 , b ] , m ∈ (0 , , and t ∈ [0 , . Obviously, if putting m = 1 in Definition 1, then f is just the ordinary logarith-mically convex on [0 , b ]. Definition 2.
A function f : [0 , b ] → (0 , ∞ ) is said to be ( α, m ) -logarithmicallyconvex if (1.2) f ( tx + m (1 − t ) y ) ≤ [ f ( x )] t α [ f ( y )] m (1 − t α ) holds for all x, y ∈ [0 , b ] , ( α, m ) ∈ (0 , × (0 , , and t ∈ [0 , . Clearly, when taking α = 1 in Definition 2, then f becomes the standard m -logarithmically convex function on [0 , b ].In [2], authors proved that the following inequalities of Hermite-Hadamard typehold for log -convex functions: Theorem 1.
Let f : I → [0 , ∞ ) be a log-convex mapping on I and a, b ∈ I with a < b . Then one has the inequality: (1.3) f ( A ( a, b )) ≤ b − a Z ba G ( f ( x ) , f ( a + b − x )) dx ≤ G ( f ( a ) , f ( b )) . Mathematics Subject Classification.
Key words and phrases.
Hadamard’s inequality, m - and ( α, m )-logarithmically convex func-tions . Theorem 2.
Let f : I → (0 , ∞ ) be a log-convex mapping on I and a, b ∈ I with a < b . Then one has the inequality: f (cid:18) a + b (cid:19) ≤ exp " b − a Z ba ln [ f ( x )] dx (1.4) ≤ b − a Z ba G ( f ( x ) , f ( a + b − x )) dx ≤ b − a Z ba f ( x ) dx ≤ L ( f ( a ) , f ( b )) ≤ f ( a ) + f ( b )2 where G ( p, q ) := √ pq is the geometric mean and L ( p, q ) := p - q ln p − ln q ( p = q ) is thelogarithmic mean of the strictly positive real numbers p, q, i.e., L ( p, q ) = p − q ln p − ln q if p = q and L ( p, p ) = p. The main purpose of this paper is to prove some inequalities of Hadamard typefor m - and ( α, m )-logarithmically convex functions. Also we give some results forlogarithmically convex functions.2. mean result We now consider the following means will be used in this paper.a) The arithmetic mean: A = A ( a, b ) := a + b , a, b ≥ , b) The geometric mean: G = G ( a, b ) := √ ab, a, b ≥ , c) The logarithmic mean: L = L ( a, b ) := (cid:26) a if a = b b − a ln b − ln a if a = b , a, b ≥ . In this section, some Hadamard type inequalities for m - and ( α, m )-logarithmicallyconvex functions will be given. Theorem 3.
Let f : [0 , ∞ ) → (0 , ∞ ) be a m -logarithmically convex function on (cid:2) , bm (cid:3) with a, b ∈ [0 , ∞ ) , a < b, m ∈ (0 , . Then (2.1) 1 b − a Z ba f ( x ) dx ≤ min { L (cid:18) f ( a ) , f (cid:18) bm (cid:19) m (cid:19) , L (cid:16) f ( b ) , f (cid:16) am (cid:17) m (cid:17) } , where L ( a, b ) is logarithmic mean.Proof. Since f is m -logarithmically convex function on (cid:2) , bm (cid:3) , we have that(2.2) f (cid:18) ta + m (1 − t ) bm (cid:19) ≤ [ f ( a )] t (cid:20) f (cid:18) bm (cid:19)(cid:21) m (1 − t ) and(2.3) f (cid:16) tb + m (1 − t ) am (cid:17) ≤ [ f ( b )] t h f (cid:16) am (cid:17)i m (1 − t )N SOME INEQUALITIES OF HADAMARD TYPE 3 for all t ∈ [0 , . By integrating the resulting inequality on [0 ,
1] with respect to t ,we get Z f ( ta + (1 − t ) b ) dt ≤ f (cid:18) bm (cid:19) m Z f ( a ) f (cid:0) bm (cid:1) ! t dt, Z f ( tb + (1 − t ) a ) dt ≤ f (cid:16) am (cid:17) m Z f ( b ) f (cid:0) am (cid:1) ! t dt. However, Z f ( a ) f (cid:0) bm (cid:1) ! t dt = f ( a ) f ( bm ) m − f ( a ) − ln f (cid:0) bm (cid:1) m = f ( a ) − f (cid:0) bm (cid:1) m f (cid:0) bm (cid:1) m h ln f ( a ) − ln f (cid:0) bm (cid:1) m i = 1 f (cid:0) bm (cid:1) m L (cid:18) f ( a ) , f (cid:18) bm (cid:19) m (cid:19) , Z f ( a ) f (cid:0) bm (cid:1) m ! t dt = 1 f (cid:0) am (cid:1) m L (cid:16) f ( b ) , f (cid:16) am (cid:17) m (cid:17) and Z f ( ta + (1 − t ) b ) dt = Z f ( tb + (1 − t ) a ) dt = 1 b − a Z ba f ( x ) dx and the inequality (2.1) is obtained. Which completes the proof. (cid:3) Theorem 4.
Let f : [0 , ∞ ) → (0 , ∞ ) be a m -logarithmically convex function on (cid:2) , bm (cid:3) with a, b ∈ [0 , ∞ ) , a < b, m ∈ (0 , . Then (2.4) f (cid:18) a + b (cid:19) ≤ b − a Z ba G (cid:18) f ( x ) , (cid:20) f (cid:18) a + b − xm (cid:19)(cid:21) m (cid:19) and (2.5) 1 b − a Z G ( f ( x ) , f ( a + b − x )) dx ≤ L (cid:18) f ( a ) f ( b ) , (cid:20) f (cid:16) am (cid:17) f (cid:18) bm (cid:19)(cid:21) m (cid:19) where G ( a, b ) , L ( a, b ) are geometric, logarithmic mean respectively.Proof. Since f is m -log-convex, we have that f (cid:18) ta + m (1 − t ) bm (cid:19) ≤ [ f ( a )] t (cid:20) f (cid:18) bm (cid:19)(cid:21) m (1 − t ) , (2.6) f (cid:16) tb + m (1 − t ) am (cid:17) ≤ [ f ( b )] t h f (cid:16) am (cid:17)i m (1 − t ) for all t ∈ [0 , G ( f ( ta + (1 − t ) b ) , f ( tb + (1 − t ) a )) ≤ [ f ( a ) f ( b )] t (cid:20) f (cid:16) am (cid:17) f (cid:18) bm (cid:19)(cid:21) m (1 − t ) MEVL¨UT TUNC¸ for all t ∈ [0 , ,
1] with respect to t , we obtain Z G ( f ( ta + (1 − t ) b ) , f ( tb + (1 − t ) a )) dt ≤ (cid:20) f (cid:16) am (cid:17) f (cid:18) bm (cid:19)(cid:21) m Z " f ( a ) f ( b ) (cid:2) f (cid:0) am (cid:1) f (cid:0) bm (cid:1)(cid:3) m t dt = (cid:20) f (cid:16) am (cid:17) f (cid:18) bm (cid:19)(cid:21) m f ( a ) f ( b ) [ f ( am ) f ( bm )] m − f ( a ) f ( b ) [ f ( am ) f ( bm )] m = f ( a ) f ( b ) − (cid:2) f (cid:0) am (cid:1) f (cid:0) bm (cid:1)(cid:3) m ln f ( a ) f ( b ) − ln (cid:2) f (cid:0) am (cid:1) f (cid:0) bm (cid:1)(cid:3) m = L (cid:18) f ( a ) f ( b ) , (cid:20) f (cid:16) am (cid:17) f (cid:18) bm (cid:19)(cid:21) m (cid:19) If we change the variable x = ta + (1 − t ) b , t ∈ [0 , Z G ( f ( ta + (1 − t ) b ) , f ( tb + (1 − t ) a )) dt = 1 b − a Z G ( f ( x ) , f ( a + b − x )) dx and the inequality in (2.5) is proved.By (2.6), for t = , we have that f (cid:18) x + y (cid:19) ≤ r [ f ( x )] h f (cid:16) ym (cid:17)i m for all x, y ∈ [0 , ∞ ). If we choose x = ta + (1 − t ) b , y = tb + (1 − t ) a , we get theinequality f (cid:18) a + b (cid:19) ≤ s [ f ( ta + (1 − t ) b )] (cid:20) f (cid:18) ta + (1 − t ) bm (cid:19)(cid:21) m for all t ∈ [0 , ,
1] over t , we obtain the inequalityin (2.4) . This completes the proof of the theorem. (cid:3) Remark 1.
If we take m = 1 in inequality (2.4) and (2.5), we obtain one inequalitysuch that special version of inequality (1.3). Theorem 5.
Let f : [0 , ∞ ) → (0 , ∞ ) be an ( α, m ) -logarithmically convex functionon (cid:2) , bm (cid:3) with a, b ∈ [0 , ∞ ) , ≤ a < b < ∞ , ( α, m ) ∈ (0 , × (0 , . Then (2.7) 1 b − a Z ba f ( x ) dx ≤ min { f (cid:18) bm (cid:19) m M ( α ) , f (cid:16) am (cid:17) m T ( α ) } , where ϕ = f ( a ) f ( bm ) m , ℓ = f ( b ) f ( am ) m and (2.8) M ( α ) = (cid:26) , ϕ = 1 ϕ α − α ln ϕ , < ϕ < and T ( α ) = (cid:26) , ℓ = 1 ℓ α − α ln ℓ , < ℓ < N SOME INEQUALITIES OF HADAMARD TYPE 5
Proof.
Since f is an ( α, m )-logarithmically convex function on (cid:2) , bm (cid:3) , we have that f (cid:18) ta + m (1 − t ) bm (cid:19) ≤ [ f ( a )] t α (cid:20) f (cid:18) bm (cid:19)(cid:21) m (1 − t α ) and f (cid:16) tb + m (1 − t ) am (cid:17) ≤ [ f ( b )] t α h f (cid:16) am (cid:17)i m (1 − t α ) for all t ∈ [0 , . Integrating the above inequality on [0 ,
1] with respect to t , we get Z f (cid:18) ta + m (1 − t ) bm (cid:19) dt ≤ f (cid:18) bm (cid:19) m Z f ( a ) f (cid:0) bm (cid:1) ! t α dt, Z f (cid:16) tb + m (1 − t ) am (cid:17) dt ≤ f (cid:16) am (cid:17) m Z f ( b ) f (cid:0) am (cid:1) ! t α dt. When ϕ = 1, we have Z ϕ t α dt = 1 . When ϕ <
1, we have Z ϕ t α dt ≤ Z ϕ αt dt = ϕ α − α ln ϕ . When ℓ = 1, we have Z ℓ t α dt = 1 . When ℓ <
1, we have Z ℓ t α dt ≤ Z ℓ αt dt = ℓ α − α ln ℓ . and Z f ( ta + (1 − t ) b ) dt = Z f ( tb + (1 − t ) a ) dt = 1 b − a Z ba f ( x ) dx and the inequality (2.7) is obtained. Which is required. (cid:3) Corollary 1.
Let f : [0 , ∞ ) → (0 , ∞ ) be an m -logarithmically convex function on (cid:2) , bm (cid:3) with a, b ∈ [0 , ∞ ) , ≤ a < b < ∞ , m ∈ (0 , . Then (2.9) 1 b − a Z ba f ( x ) dx ≤ min { f (cid:18) bm (cid:19) m M (1) , f (cid:16) am (cid:17) m T (1) } , where M, T are defined as in (2.8). At the same time, this inequality is the sameas inequality (2.1).
Theorem 6.
Let f : [0 , ∞ ) → (0 , ∞ ) be an ( α, m ) -logarithmically convex functionon (cid:2) , bm (cid:3) with a, b ∈ [0 , ∞ ) , ≤ a < b < ∞ , ( α, m ) ∈ (0 , × (0 , . Then (2.10) 1 b − a Z G ( f ( x ) , f ( a + b − x )) dx ≤ (cid:20) f (cid:16) am (cid:17) f (cid:18) bm (cid:19)(cid:21) m S ( α ) where θ = f ( a ) f ( b ) h f (cid:16) am (cid:17)i − m (cid:20) f (cid:18) bm (cid:19)(cid:21) − m MEVL¨UT TUNC¸ and S ( α ) = (cid:26) , θ = 1 , θ α − α ln θ , < θ < ,G ( a, b ) , L ( a, b ) are geometric, logarithmic mean respectively.Proof. Since f is m -log-convex, we have that f (cid:18) ta + m (1 − t ) bm (cid:19) ≤ [ f ( a )] t α (cid:20) f (cid:18) bm (cid:19)(cid:21) m (1 − t α ) , (2.11) f (cid:16) tb + m (1 − t ) am (cid:17) ≤ [ f ( b )] t α h f (cid:16) am (cid:17)i m (1 − t α ) for all t ∈ [0 , G ( f ( ta + (1 − t ) b ) , f ( tb + (1 − t ) a )) ≤ [ f ( a ) f ( b )] t α (cid:20) f (cid:16) am (cid:17) f (cid:18) bm (cid:19)(cid:21) m (1 − t α ) = (cid:20) f (cid:16) am (cid:17) f (cid:18) bm (cid:19)(cid:21) m θ t α for all t ∈ [0 , ,
1] with respect to t , we obtain Z G ( f ( ta + (1 − t ) b ) , f ( tb + (1 − t ) a )) dt ≤ (cid:20) f (cid:16) am (cid:17) f (cid:18) bm (cid:19)(cid:21) m Z θ t α dt When θ = 1, we have R θ t α dt = 1 . When θ <
1, we have Z θ t α dt ≤ Z θ αt dt = θ α − α ln θ . If we change the variable x = ta + (1 − t ) b , t ∈ [0 , Z G ( f ( ta + (1 − t ) b ) , f ( tb + (1 − t ) a )) dt = 1 b − a Z G ( f ( x ) , f ( a + b − x )) dx and the inequality in (2.10) is proved. (cid:3) References [1] R.-F. Bai, F. Qi and B.-Y. Xi, and : Hermite-Hadamard type inequalities for the m - and( α, m )-logarithmically convex functions. Filomat 27 (2013), 1-7.[2] Dragomir, S.S. and Mond, B.: Integral inequalities of Hadamard type for log-convex functions,Demonstratio Mathematica, 31 (2) (1998), 354-364. Kilis 7 Aralık University, Faculty of Science and Arts, Department of Mathematics,Kilis, 79000, Turkey.
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