Vol. 16 (2015), No. 1, pp. 491–501
SOME NEW INEQUALITIES OF HERMITE-HADAMARD TYPE
FOR s-CONVEX FUNCTIONS
MEHMET ZEKI SARIKAYA AND MEHMET EY ¨UP KIRIS
Received 15 January, 2014
Abstract. Some new results related of the left-hand side of the Hermite-Hadamard type inequal-ities for the class of mappings whose second derivatives at certain powers are s convex in the second sense are established. Also, some applications to special means of real numbers are provided.
2010 Mathematics Subject Classification: 26A51; 26D07; 26D10; 26D15
Keywords: Hermite-Hadamard type inequality, s convex function, H¨older inequality
1. INTRODUCTION
Let f W I R ! R be a convex mapping defined on the interval I of real num-bers and a; b2 I with a < b. The following double inequality is well known in the literature as Hermite-Hadamard inequality [6]:
f a C b 2 1 b a Z b a f .x/dx f .a/C f .b/ 2 :
Both inequalities hold in the reversed direction if f is concave. For recent results, generalizations and new inequalities related to the Hermite-Hadamard inequality see [3,4,9,11,12,14,17].
The classical Hermite- Hadamard inequality provides estimates of the mean value of a continuous convex function f W Œa; b ! R.
Definition 1. Let I be on interval in R. Then f W I ! R is said to be convex if the following inequality holds
f .xC .1 /y/ f .x/ C .1 /f .y/
for all x; y2 I and 2 Œ0; 1 . We say that f is concave if . f / is convex.
The class of functions which are s-convex in the second sense has been stated as the following (see [7]).
c
Definition 2. Let s be a real number, s2 .0; 1 . A function f W Œ0; 1/ ! R is said to be s-convex (in the second sense), if
f .xC .1 /y/ sf .x/C .1 /sf .y/
for all x; y2 Œ0; 1/ and 2 Œ0; 1. Some interesting and important inequalities for s-convex (in the second sense) functions can be found in [1,2,5,8,10,15,16,18,19]. It can be easily seen that convexity means just s-convexity when sD 1.
In [13], Sarikaya et. al. established inequalities for twice differentiable convex mappings which are connected with Hadamard’s inequality, and they used the fol-lowing lemma to prove their results:
Lemma 1. Letf W I R ! R be twice differentiable function on Iı,a; b2 Iı witha < b: If f002 L1Œa; b, then
1 b a Z b a f .x/dx f a C b 2 D.b a/ 2 2 Z 1 0 m .t /f00.t aC .1 t/b/ C f00.t bC .1 t/a/ dt; where m.t /WD 8 < : t2 ; t2 Œ0;12/ .1 t /2 ; t2 Œ12; 1:
Also, the main inequalities in [13], pointed out as follows:
Theorem 1. Letf W I R ! R be twice differentiable function on Iıwithf002 L1Œa; b. Ifjf00j is convex on Œa; b; then
ˇ ˇ ˇ ˇ ˇ 1 b a Z b a f .x/dx f a C b 2 ˇˇ ˇ ˇ ˇ .b a/ 2 24 jf00.a/j C jf00.b/j 2 : (1.1)
Theorem 2. Letf W I R ! R be twice differentiable function on Iısuch that f002 L1Œa; b where a; b2 I; a < b. If jf00jqis convex onŒa; b; q > 1, then
ˇ ˇ ˇ ˇ ˇ 1 b a Z b a f .x/dx f a C b 2 ˇˇ ˇ ˇ ˇ .b a/ 2 8 .2pC 1/1=p jf00.a/jqC jf00.b/jq 2 1=q (1.2) where p1C1q D 1:
The main aim of this paper is to establish new inequalities of Hermite-Hadamard type for the class of functions whose second derivatives at certain powers are s-convex functions in the second sense.
2. MAIN RESULTS
In order to prove our main results we need the following lemma:
Lemma 2. Letf W I R ! R be twice differentiable function on Iı,a; b2 Iı witha < b. If f002 L Œa; b, then the following equaliy holds:
1 b a Z b a f .x/dx f a C b 2 D.b a/ 2 16 Z 1 0 t2f00 t 2aC 2 t 2 b dtC Z 1 0 t2f00 2 t 2 aC t 2b dt : (2.1) Proof. By integration by parts, we have the following identity
Z 1 0 t2f00 t 2aC 2 t 2 b dtC Z 1 0 t2f00 2 t 2 aC t 2b dt D 4 b a Z 1 0 tf0 t 2aC 2 t 2 b dt Z 1 0 tf0 2 t 2 aC t 2b dt D 4 b a ( t 2 a bf t 2aC 2 t 2 b ˇ ˇ ˇ ˇ 1 0 C 2 b a Z 1 0 f t 2aC 2 t 2 b dt C tb2af 2 t 2 aC t 2b ˇ ˇ ˇ ˇ 1 0C 2 b a Z 1 0 f 2 t 2 aC t 2b dt ) D 16 .b a/2f a C b 2 C 8 .b a/2 Z 1 0 f t 2aC 2 t 2 b dt C 8 .b a/2 Z 1 0 f 2 t 2 aC t 2b dt:
Using the change of the variable in last integrals, we get the required identity (2.1). Theorem 3. Letf W I Œ0; 1/ ! R be twice differantiable mapping on Iı,a; b2 I with a < b . If jf00j is s- convex in the second sense on Œa; b, for some fixed s2 .0; 1, then the following inequaliy holds:
ˇ ˇ ˇ ˇ ˇ 1 b a Z b a f .x/dx f a C b 2 ˇˇ ˇ ˇ ˇ .b a/ 2 .2sC4 .4sC 12// 2sC4.sC 1/ .s C 2/ .s C 3/ ˇ ˇf00.a/ˇˇC ˇ ˇf00.b/ˇˇ : (2.2) Proof. Using Lemma2and the s- convexity in the second sense ofjf00j, we find
ˇ ˇ ˇ ˇ ˇ 1 b a Z b a f .x/dx f a C b 2 ˇˇ ˇ ˇ ˇ
.b a/ 2 16 Z 1 0 t2 ˇ ˇ ˇ ˇ f00 t 2aC 2 t 2 b ˇ ˇ ˇ ˇC ˇ ˇ ˇ ˇ f00 2 t 2 aC t 2b ˇ ˇ ˇ ˇ dt .b a/ 2 16 Z 1 0 t2 t 2 s ˇ ˇf00.a/ˇˇC 2 t 2 s ˇ ˇf00.b/ˇˇ C 2 t 2 s ˇ ˇf00.a/ ˇ ˇC t 2 s ˇ ˇf00.b/ ˇ ˇ dt D.b a/ 2. jf00.a/j C jf00.b/j/ 2sC4 Z 1 0 tsC2C t2.2 t /s dt D.b a/ 2 .2sC4 .4sC 12// 2sC4.sC 1/ .s C 2/ .s C 3/ ˇ ˇf00.a/ˇˇC ˇ ˇf00.b/ˇˇ
where we have used the fact that Z 1 0 t2.2 t /sdt D2 sC4 s2 C 7s C 14 .sC 1/ .s C 2/ .s C 3/; and Z 1 0 tsC2dtD 1 sC 3:
The proof is completed.
Remark1. If we take sD 1 in Theorem3, then inequality (2.2) becomes inequality (1.1).
The next theorem gives a new upper bound of the left-hand side Hermite-Hadamard inequality for s-convex functions:
Theorem 4. Let f W I Œ0; 1/ ! R be twice differantiable mapping on Iı , a; b2 I with a < b. If jf00jq iss- convex in the second sense on Œa; b , for some fixeds2 .0; 1 and q > 1, then the following inequaliy holds:
ˇ ˇ ˇ ˇ ˇ 1 b a Z b a f .x/dx f a C b 2 ˇˇ ˇ ˇ ˇ .b a/ 2 2sC4 2s 2pC 1 p1 2 4 jf00.a/jqC 2sC1 1 jf00.b/jq sC 1 !q1 C 2 sC1 1 jf00.a/jq C jf00.b/jq sC 1 !q13 5 (2.3) .b a/ 2 8 2s 2pC 1 p1 1 .sC 1/1q ˇ ˇf00.a/ˇˇC ˇ ˇf00.b/ˇˇ ; where p1C1q D 1:
Proof. From Lemma2, using H¨older’s inequality and the s-convexity in the second sense ofjf00jq, we find ˇ ˇ ˇ ˇ ˇ 1 b a Z b a f .x/dx f a C b 2 ˇˇ ˇ ˇ ˇ .b a/ 2 16 Z 1 0 jtj 2p dt 1 p 8 < : Z 1 0 ˇ ˇ ˇ ˇ f00 t 2aC 2 t 2 b ˇ ˇ ˇ ˇ q dt 1 q C Z 1 0 ˇ ˇ ˇ ˇ f00 2 t 2 aC t 2b ˇ ˇ ˇ ˇ q dt 1 q 9 = ; .b a/ 2 16 1 2pC 1 p1 8 < : Z 1 0 t 2 s ˇ ˇf00.a/ ˇ ˇ q C 2 t 2 s ˇ ˇf00.b/ ˇ ˇ q dt 1 q C Z 1 0 2 t 2 s ˇ ˇf00.a/ˇˇ q C t2 s ˇ ˇf00.b/ˇˇ q dt 1 q 9 = ; .b a/ 2 2sC4 2s 2pC 1 p1 8 < : jf00.a/jqC 2sC1 1 jf00.b/jq sC 1 !1q C 2 sC1 1 jf00.a/jq C jf00.b/jq sC 1 !1q9 = ; :
Let a1 D jf00.a/jq; b1 D 2sC1 1 jf00.b/jq; a2D 2sC1 1 jf00.a/jq; b2D
jf00.b/jq. Here , 0 <q1< 1 for q > 1. Using the fact that
n X kD1 .akC bk/s n X kD1 askC n X kD1 bsk: for .0 s < 1/ ; a1; a2; :::; an 0; b1; b2; :::; bn 0; we obtain ˇ ˇ ˇ ˇ ˇ 1 b a Z b a f .x/dx f a C b 2 ˇˇ ˇ ˇ ˇ .b a/ 2 2sC4 2s 2pC 1 p1 1 sC 1 1q ˇ ˇf00.a/ ˇ ˇC 2sC1 1 1qˇ ˇf00.b/ ˇ ˇ C 2sC1 1 1 qˇ ˇf00.a/ ˇ ˇC ˇ ˇf00.b/ ˇ ˇ D.b a/ 2 2sC4 2s 2pC 1 p1 1 sC 1 q1 1C 2sC1 1 1 q ˇ ˇf00.a/ˇˇC ˇ ˇf00.b/ˇˇ .b a/ 2 2sC4 2s 2pC 1 p1 1 sC 1 1q 2sC1 ˇ ˇf00.a/ˇˇC ˇ ˇf00.b/ˇˇ :
Corollary 1. Under assumption Theorem4, if we takesD 1, then the inequality (2.3) becomes the following inequality
ˇ ˇ ˇ ˇ ˇ 1 b a Z b a f .x/dx f a C b 2 ˇˇ ˇ ˇ ˇ .b a/ 2 16 .2pC 1/p1 2 4 jf00.a/jq C 3 jf00.b/jq 4 1q C 3 jf 00.a/jq C jf00.b/jq 4 1q 3 5 .b a/ 2 22qC2.2pC 1/ 1 p ˇ ˇf00.a/ˇˇC ˇ ˇf00.b/ˇˇ :
Theorem 5. Let f W I Œ0; 1/ ! R be twice differantiable mapping on Iı , a; b2 I with a < b. If jf00jqiss-convex in the second sense on Œa; b , for some fixed s2 .0; 1 and q 1, then the following inequaliy holds:
ˇ ˇ ˇ ˇ ˇ 1 b a Z b a f .x/dx f a C b 2 ˇˇ ˇ ˇ ˇ (2.4) .b a/ 2 16 1 3 1 1q 2 4 1 2s.sC 3/ ˇ ˇf00.a/ˇˇ q C 2 sC4 s2 C 7s C 14 2s.sC 1/ .s C 2/ .s C 3/ ˇ ˇf00.b/ˇˇ q !1q C 2 sC4 s2 C 7s C 14 2s.sC 1/ .s C 2/ .s C 3/ ˇ ˇf00.a/ˇˇ q C 1 2s.sC 3/ ˇ ˇf00.b/ˇˇ q !1q3 5:
Proof. From Lemma 2, using the well known power mean inequality for q 1 and the s- convexity in the second sense ofjf00jq, we find
ˇ ˇ ˇ ˇ ˇ 1 b a Z b a f .x/dx f a C b 2 ˇˇ ˇ ˇ ˇ .b a/ 2 16 8 < : Z 1 0 t2 1 1 q 2 4 Z 1 0 t2 ˇ ˇ ˇ ˇ f00 t 2aC 2 t 2 b ˇ ˇ ˇ ˇ q dt 1 q C Z 1 0 t2 ˇ ˇ ˇ ˇ f00 2 t 2 aC t 2b ˇ ˇ ˇ ˇ q dt 1 q 3 5 9 = ; .b a/ 2 16 1 3 1 1q 2 4 Z 1 0 t2 t 2 s ˇ ˇf00.a/ˇˇ q C 2 t 2 s ˇ ˇf00.b/ˇˇ q dt 1 q
C Z 1 0 t2 2 t 2 s ˇ ˇf00.a/ˇˇ q C t 2 s ˇ ˇf00.b/ˇˇ q dt 1 q 3 5 .b a/ 2 16 1 3 1 1q 2 4 1 2s.sC 3/ ˇ ˇf00.a/ˇˇ q C 2 sC4 s2 C 7s C 14 2s.sC 1/ .s C 2/ .s C 3/ ˇ ˇf00.b/ˇˇ q !1q C 2 sC4 s2 C 7s C 14 2s.sC 1/ .s C 2/ .s C 3/ ˇ ˇf00.a/ ˇ ˇ q C 1 2s.sC 3/ ˇ ˇf00.b/ ˇ ˇ q !1q3 5
which completes the proof of Theorem5.
Corollary 2. Under assumption Theorem5, if we takesD 1, then inequality (2.4) becomes the following inequality:
ˇ ˇ ˇ ˇ ˇ 1 b a Z b a f .x/dx f a C b 2 ˇˇ ˇ ˇ ˇ .b a/ 2 48 2 4 3 jf00.a/jqC 5 jf00.b/jq 8 1 q C 5 jf 00.a/jq C 3 jf00.b/jq 8 1 q 3 5:
3. APPLICATIONS TO SPECIAL MEANS
In [1], the following result is given.
Let gW I ! I1 Œ0; 1/ be a non-negative convex functions on I: Then gs.x/ is
s convex on I; 0 < s < 1:
For arbitrary positive real numbers a; b .a¤ b/, we shall consider the following special means:
(a) The arithmetic mean: AD A.a; b/ WDaC b
2 ; a; b > 0; (b) The harmonic mean:
H D H .a; b/ WD 2ab
aC b; a; b > 0; (c) The logarithmic mean:
LD L .a; b/ WD 8 < : a if aD b b a ln b ln a if a¤ b , a; b > 0;
LpD Lp.a; b/WD 8 ˆ < ˆ : h bpC1 apC1 .pC1/.b a/ ip1 if a¤ b a if aD b , p2 RŸ f 1; 0g I a; b > 0.
It is well known that Lp is monotonic nondecreasing over p2 R with L 1WD L
and L0WD I: In particular, we have the following inequalities
H L A:
Now, using the results of Section2, some new inequalities is derived for the above means.
(1) Let f W Œa; b ! R, .0 < a < b/, f .x/ D xsC1; s2 .0; 1. Then, 1 b a Z b a f .x/dxD LsC1sC1.a; b/ ; f .a/C f .b/ 2 D A.a sC1; bsC1/; f a C b 2 D AsC1.a; b/: (a) From Theorem3, we obtain
ˇ ˇLsC1sC1.a; b/ AsC1.a; b/ ˇ ˇ .b a/2s.2sC4 .4sC 12// 2sC3.sC 2/ .s C 3/ A.a s 1 ; bs 1/: For instance, if sD 1 then we get
ˇ ˇL22.a; b/ A2.a; b/ˇ ˇ 1 12.b a/ 2:
(b) From Theorem4, we have ˇ ˇLsC1sC1.a; b/ AsC1.a; b/ˇˇ s .b a/ 2 2sC4 2s 2pC 1 p1 a.s 1/qC 2sC1 1 b.s 1/q 1 q C 2sC1 1 a.s 1/qC b.s 1/q 1 q s .b a/ 2 4 2s 2pC 1 p1 1 .sC 1/1q 1 A.as 1; bs 1/;
where q > 1 and p1Cq1D 1: For instance, if s D 1 then we have
ˇ ˇL22.a; b/ A2.a; b/ˇˇ .b a/2 4 1 4pC 2 p1 ; p > 1:
(c) From Theorem5, we get ˇ ˇLsC1sC1.a; b/ AsC1.a; b/ˇˇ .b a/2 2sC4.sC 2/ .s C 3/ 1 3 1 q1 s.sC 1/.s C 2/a.s 1/qC 2sC4s s3 7s2 14s b.s 1/q 1 q C 2sC4s s3 7s2 14s a.s 1/qC s.s C 1/.s C 2/b.s 1/q 1 q
where q > 1 and p1Cq1D 1: For instance, if s D 1 then we have
ˇ ˇL22.a; b/ A2.a; b/ˇˇ .b a/2.48/1q 576 ; q > 1: (2) Let f W Œa; b Œ0; 1/ ! R, .0 < a < b/, f .x/ D 1 xs 2 K 2 s; s2 .0; 1. Then, 1 b a Z b a f .x/dxD L ss.a; b/ ; f .a/C f .b/ 2 D A.a s; b s/; f .aC b 2 /D A s.a; b/:
(a) From Theorem3, we obtain
jL ss.a; b/ A s.a; b/j
.b a/2.2sC4 .4sC 12// 2sC4.sC 1/ .s C 2/ .s C 3/
h
a. s 2/C b. s 2/i: For instance, if sD 1 then we get
ˇ ˇL 11.a; b/ A 1.a; b/ ˇ ˇ .b a/2 24 A a 3 ; b 3 : (b) From Theorem4, we have
jL ss.a; b/ A s.a; b/j s .b a/ 2 2sC4 2s 2pC 1 p1 a. s 2/qC 2sC1 1 b. s 2/q 1 q C 2sC1 1 a. s 2/qC b. s 2/q 1 q .b a/ 2 8 2s 2pC 1 p1 s .sC 1/q1 1 h a. s 2/C b. s 2/i;
where q > 1 and p1Cq1D 1: For instance, if s D 1 then we have ˇ ˇL 11.a; b/ A 1.a; b/ ˇ ˇ .b a/ 2 32 2 2pC 1 p1 a 3qC 3b 3q1q C 3a 3qC b 3qq1 .b a/ 2 8 4 2pC 1 p1 A a 3; b 3 where q > 1:
(c) From Theorem5, we get
jL ss.a; b/ A s.a; b/j .b a/2 2sC4.sC 2/ .s C 3/ 1 3 1 1q s.sC 1/.s C 2/a. s 2/qC 2sC4s s3 7s2 14s b. s 2/q 1 q C 2sC4s s3 7s2 14s a. s 2/qC s.s C 1/.s C 2/b. s 2/q 1 q
where q > 1 and p1Cq1D 1: For instance, if s D 1 then we have
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Authors’ addresses
Mehmet Zeki Sarıkaya
Department of Mathematics, Faculty of Science and Arts, D¨uzce University, D¨uzce, Turkey E-mail address: sarikayamz@gmail.com
Mehmet Ey ¨up Kiris
Department of Mathematics, Faculty of Science and Arts, Afyon Kocatepe University, Afyon, Tur-key
