ON HERMITE-HADAMARD TYPE INEQUALITIES FOR F-CONVEX FUNCTION

Miskolc Mathematical Notes HU e-ISSN 1787-2413 Vol. 20 (2019), No. 1, pp. 169–191 DOI: 10.18514/MMN.2019.2436

ON HERMITE-HADAMARD TYPE INEQUALITIES FOR F -CONVEX FUNCTION

H. BUDAK, T. TUNC¸ , AND M. Z. SARIKAYA

Received 27 October, 2017

Abstract. In this study, we firstly give some properties the family F and F -convex function which are defined by B. Samet. Then, we obtain some midpoint inequalities for differentiable function. Moreover, we establish some midpoint and trapezoid type inequalities for function whose second derivatives in absolute value are F -convex.

2010 Mathematics Subject Classification: 26D07; 26D10; 26D15; 26A33

Keywords: Hermite-Hadamard inequality, F convex, midpoint inequality, trapezoid inequality

1. INTRODUCTION

Let f W I  R ! R be a convex function on the interval I of real numbers and a; b2 I with a < b. If f is a convex function then the following double inequality, which is well known in the literature as the Hermite–Hadamard inequality, holds [8]

f a C b 2   1 b a Z b a f .x/dx_ f .a/C f .b/ 2 : (1.1)

Note that some of the classical inequalities for means can be derived from (1.1) for appropriate particular selections of the mapping f . Both inequalities hold in the reversed direction if f is concave (1.1).

It is well known that the Hermite–Hadamard inequality plays an important role in nonlinear analysis. Over the last decade, this classical inequality has been improved and generalized in a number of ways; there have been a large number of research papers written on this subject, (see, [3,4,6,9,12,13]) and the references therein.

Over the years, many type of convexity have been defined, such as quasi-convex [1], pseudo-convex [7], strongly convex [10], " convex [2], s convex [5], h convex [14] and etc. Recently, Samet [11] have defined a new concept of convexity that de-pends on a certain function satisfying some axioms, that generalizes different types of convexity, including " convex functions, ˛ convex functions, h convex functions and many others.

Recall the familyF of mappings F W R  R  R Œ0; 1 ! R satisfying the follow-ing axioms:

c

(A1) If ui2 L1.0; 1/; iD 1; 2; 3; then for every  2 Œ0; 1 ; we have 1 Z 0 F .u1.t /; u2.t /; u3.t /; /dtD F 0 @ 1 Z 0 u1.t /dt; 1 Z 0 u2.t /dt; 1 Z 0 u3.t /dt;  1 A:

(A2) For every u2 L1.0; 1/ ; w_{2 L}1.0; 1/ and .´1; ´2/2 R2; we have 1 Z 0 F .w.t /u.t /; w.t /´1; w.t /´2; t /dtD TF ;w 0 @ 1 Z 0 w.t /u.t /dt; ´1; ´2/ 1 A; where TF ;w W R  R  R ! R is a function that depends on (F; w), and it is

non-decreasing with respect to the first variable.

(A3) For any .w; u1; u2; u3/2 R4; u42 Œ0; 1 ; we have

wF .u1; u2; u3; u4/D F .wu1; wu2; wu3; u4/C Lw

where Lw2 R is a constant that depends only on w:

Definition 1. Let f W Œa; b ! R; .a; b/ 2 R2; a < b, be a given function. We say that f is a convex function with respect to some F 2 F (or F convex function) iff

F .f .tx_{C .1 t/y/; f .x/; f .y/; t/  0; .x; y; t/ 2 Œa; b  Œa; b  Œ0; 1 :} Remark1. 1) Let " 0; and let f W Œa; b ! R, .a; b/ 2 R2; a < b; be an " convex function, that is (see [2])

f .tx_{C .1 t/y/  tf .x/ C .1 t/f .y/ C "; .x; y; t/ 2 Œa; b  Œa; b  Œ0; 1 :} Define the functions F W R  R  R Œ0; 1 ! R by

F .u1; u2; u3; u4/D u1 u4u2 .1 u4/u3 " (1.2) and TF ;wW R  R  R ! R by TF ;w.u1; u2; u3/D u1 0 @ 1 Z 0 t w.t /dt 1 Au2 0 @ 1 Z 0 .1 t /w.t /dt 1 Au3 ": (1.3) For LwD .1 w/"; (1.4)

it is clear that F 2 F and

F .f .tx_{C.1 t/y/; f .x/; f .y/; t/ D f .tx C.1 t/y/ tf .x/ .1 t/f .y/ "  0;} that is f is an F convex function. Particularly, taking "D 0; we show that if f is a convex function then f is an F convex function with respect to F defined above.

2) Let f W Œa; b ! R, .a; b/ 2 R2; a < b; be an ˛ convex function, ˛_{2 .0; 1, that} is

Define the functions F W R  R  R Œ0; 1 ! R by F .u1; u2; u3; u4/D u1 u˛4u2 .1 u˛4/u3 (1.5) and TF ;wW R  R  R ! R by TF ;w.u1; u2; u3/D u1 0 @ 1 Z 0 t˛w.t /dt 1 Au2 0 @ 1 Z 0 .1 t˛/w.t /dt 1 Au3: (1.6)

For LwD 0; it is clear that F 2 F and

F .f .txC .1 t/y/; f .x/; f .y/; t/ D f .tx C .1 t/y/ t˛f .x/ .1 t˛/f .y/ 0; that is f is an F convex function.

3) Let hW J ! Œ0; 1/ be a given function which is not identical to 0; where J is an interval in R such that .0; 1/ J: Let f W Œa; b ! Œ0; 1/, .a; b/ 2 R2; a < b; be an h convex function, that is (see [14])

f .tx_{C .1 t/y/  h.t/f .x/ C h.1 t/f .y/; .x; y; t/ 2 Œa; b  Œa; b  Œ0; 1 :} Define the functions F W R  R  R Œ0; 1 ! R by

F .u1; u2; u3; u4/D u1 h.u4/u2 h.1 u4/u3 (1.7) and TF ;wW R  R  R ! R by TF ;w.u1; u2; u3/D u1 0 @ 1 Z 0 h.t /w.t /dt 1 Au2 0 @ 1 Z 0 h.1 t /w.t /dt 1 Au3: (1.8)

For LwD 0; it is clear that F 2 F and

F .f .tx_{C.1 t/y/; f .x/; f .y/; t/ D f .tx C.1 t/y/ h.t/f .x/ h.1 t/f .y/  0;} that is f is an F convex function.

In [11], author established the following Hermite-Hadamard type inequalities us-ing the new convexity concept:

Theorem 1. Letf _{W Œa; b ! R, .a; b/ 2 R}2,a < b, be an F convex function, for someF _{2 F . Suppose that f 2 L}1Œa; b. Then

F f  a C b 2  ; 1 b a Z b a f .x/dx; 1 b a Z b a f .x/dx;1 2 !  0; TF ;1 1 b a Z b a f .x/dx; f .a/; f .b/ !  0:

Theorem 2. Let f W Iı R ! R be a differentiable mapping on Iı; .a; b/2 Iı_{ I}ı; a < b: Suppose that

(i)_jf0_{j is F convex on Œa; b ; for some F 2 F}

(ii) the functiont _{2 .0; 1/ ! L}w.t / belongs to L1.0; 1/ ; where w.t /D j1 2tj.

Then, TF ;w 2 b a ˇ ˇ ˇ ˇ ˇ f .a/_{C f .b/} 2 1 b a Z b a f .x/dx ˇ ˇ ˇ ˇ ˇ ;ˇ ˇf0.a/ˇ ˇ; ˇ ˇf0.b/ˇ ˇ ! C 1 Z 0 Lw.t /dt 0:

Theorem 3. Let f _{W I}ı_{ R ! R be a differentiable mapping on I}ı; .a; b/₂ Iı_{ I}ı; a < b and let p > 1: Suppose that_jf0_jp=.p 1/ isF convex on Œa; b ; for someF _{2 F and jf}0_{j 2 L}p=.p 1/.a; b/: Then

TF ;1  A.p; f /;ˇˇf0.a/ˇˇ p=.p 1/ ;ˇˇf0.b/ˇˇ p=.p 1/  0 where A.p; f /_D  2 b a _pp₁ .p_{C 1/}p11 ˇ ˇ ˇ ˇ ˇ f .a/C f .b/ 2 1 b a Z b a f .x/dx ˇ ˇ ˇ ˇ ˇ p p 1 : The following lemmas will be useful when we give our main result.

Lemma 1 ([6]). Let f W Iı R ! R be a differentiable mapping on Iı,a; b_{2 I}ı witha < b: If f02 L1Œa; b, then we have

1 b a Z b a f .x/dx f a C b 2  D .b a/ Z 1 0 p .t / f0.t a_{C .1 t/b/dt;} where p.t /_D 8 < : t; t20;1 2  t 1; t ₂ 1₂; 1 :

Lemma 2 ([4]). Let f W Iı R ! R be twice differentiable function on Iı,a; b2 Iıwitha < b: If f00_{2 L}1Œa; b, then

f .a/_{C f .b/} 2 1 b a Z b a f .x/dxD.b a/ 2 2 Z 1 0 t .1 t /f00.t aC .1 t/b/dt: (1.9) Lemma 3 ( [13] ). Let f W Iı R ! R be twice differentiable function on Iı, a; b2 Iıwitha < b: If f002 L1Œa; b, then

1 b a Z b a f .x/dx f  a C b 2  (1.10)

D.b a/ 2 4 Z 1 0 m .t /f00_{.t aC .1 t/b/ C f}00_{.t bC .1 t/a/ dt;} where m.t /_D 8 < : t2; t20;1 2  .1 t /2; t₂ 1₂; 1 :

2. MIDPOINT TYPE INEQUALITIES FOR DIFFERENTIABLE FUNCTIONS

In this section, we establish some midpoint type inequalities for functions whose derivatives absolute values are F convex.

Theorem 4. LetI_{ R be an interval, f W I}ı_{ R ! R be a differentiable mapping} onIı; a; b2 Iı; a < b: Suppose that_jf0_{j is F convex on Œa; b ; for some F 2 F} and the functiont_{2 Œ0; 1 ! L}w.t /belongs toL1Œ0; 1 ; where w.t /D jp.t/j (p.t/ is

defined as in Lemma1). Then,

TF ;w 1 b a ˇ ˇ ˇ ˇ ˇ 1 b a Z b a f .x/dx f  a C b 2 ˇ_ˇ ˇ ˇ ˇ ;ˇˇf0.a/ˇˇ; ˇ ˇf0.b/ˇˇ ! C 1 Z 0 Lw.t /dt 0:

Proof. Sincejf0j is F convex, we have F ˇˇf0.t a_{C .1 t/b/}ˇˇ;

ˇ

ˇf0.a/ˇˇ; ˇ

ˇf0.b/ˇˇ; t
 0; t 2 Œ0;1: Multiplying this inequality by w.t / and using axiom (A3), we get

F w.t /ˇˇf0.t a_{C .1 t/b/}ˇˇ; w.t / ˇ

ˇf0.a/ˇˇ; w.t / ˇ

ˇf0.b/ˇˇ; t
C L_{w.t /} 0; t 2 Œ0; 1 : Integrating over Œ0; 1 with respect to the variable t and using axiom (A2), we obtain

TF ;w 0 @ 1 Z 0 w.t /ˇˇf0.t aC .1 t/b/ ˇ ˇdt; ˇ ˇf0.a/ ˇ ˇ; ˇ ˇf0.b/ ˇ ˇ 1 AC 1 Z 0 Lw.t /dt 0; t 2 Œ0; 1 :

On the other hand using Lemma1we have 1 b a ˇ ˇ ˇ ˇ ˇ 1 b a Z b a f .x/dx f a C b 2 ˇ_ˇ ˇ ˇ ˇ  Z 1 0 jp.t/j ˇ ˇf0.t a_{C .1 t/b/}ˇˇdt: Since TF ;w is nondecreasing with respect to the first variable, we get

TF ;w 1 b a ˇ ˇ ˇ ˇ ˇ 1 b a Z b a f .x/dx f  a C b 2 ˇ_ˇ ˇ ˇ ˇ ;ˇˇf0.a/ˇˇ; ˇ ˇf0.b/ˇˇ ! C 1 Z 0 Lw.t /dt 0:

Corollary 1. If_jf0_{j is " convex on Œa; b ; "  0; then we have} ˇ ˇ ˇ ˇ ˇ 1 b a Z b a f .x/dx f  a C b 2 ˇ_ˇ ˇ ˇ ˇ  .b a/ jf 0_{.a/j C jf}0_.b/j 8 C " 4  : Proof. It is known that an " convex is an F convex. Using (1.4) with w.t /_{D jp.t/j ; we obtain} 1 Z 0 Lw.t /dtD " 1 Z 0 .1 _{jp.t/j/dt D}3 4": From (1.3) with w.t /D jp.t/j ; we get

TF ;w.u1; u2; u3/Du1 0 @ 1 Z 0 t_{jp.t/j dt} 1 Au2 0 @ 1 Z 0 .1 t /_{jp.t/j dt} 1 Au3 " Du1 u2C u3 8 " for u1; u2; u32 R: Hence, TF ;w 1 b a ˇ ˇ ˇ ˇ ˇ 1 b a Z b a f .x/dx f a C b 2 ˇ_ˇ ˇ ˇ ˇ ;ˇˇf0.a/ˇˇ; ˇ ˇf0.b/ˇˇ ! D 1 b a ˇ ˇ ˇ ˇ ˇ 1 b a Z b a f .x/dx f  a C b 2 ˇ_ˇ ˇ ˇ ˇ jf0.a/_{j C jf}0.b/_j 8 ":

Thus, by Theorem4, we have 1 b a ˇ ˇ ˇ ˇ ˇ 1 b a Z b a f .x/dx f  a C b 2 ˇ_ˇ ˇ ˇ ˇ jf0.a/_{j C jf}0.b/_j 8 "C 3 4" 0; that is ˇ ˇ ˇ ˇ ˇ 1 b a Z b a f .x/dx f  a C b 2 ˇ_ˇ ˇ ˇ ˇ  .b a/ jf 0_{.a/j C jf}0_.b/j 8 C " 4  :

This completes the proof. 

Remark2. If we take "D 0 in Corollary1, thenjf0j is convex and we have the inequality ˇ ˇ ˇ ˇ ˇ 1 b a Z b a f .x/dx f  a C b 2 ˇ_ˇ ˇ ˇ ˇ  .b a/ jf 0_{.a/j C jf}0_.b/j 8  which is given by [6].

Corollary 2. If_jf0_{j is ˛ convex on Œa; b ; ˛ 2 .0; 1; then we have} ˇ ˇ ˇ ˇ ˇ 1 b a Z b a f .x/dx f  a C b 2 ˇ_ˇ ˇ ˇ ˇ  .b a/  ₁ .˛_{C 1/.˛ C 2/}  1 1 2˛C1  ˇ ˇf0.a/ˇˇ C 1 4 1 .˛C 1/.˛ C 2/  1 1 2˛C1  ˇ ˇf0.b/ˇˇ  : Proof. It is known that an ˛ convex is an F convex. Using (1.6) with w.t /D jp.t/j ; we obtain TF ;w.u1; u2; u3/Du1 0 @ 1 Z 0 t˛_{jp.t/j dt} 1 Au2 0 @ 1 Z 0 .1 t˛/_{jp.t/j dt} 1 Au3 Du1 1 .˛_{C 1/.˛ C 2/}  1 1 2˛C1  u2  1 4 1 .˛_{C 1/.˛ C 2/}  1 1 2˛C1  u3

for u1; u2; u32 R: Hence, by Theorem4, we have

TF ;w 1 b a ˇ ˇ ˇ ˇ ˇ 1 b a Z b a f .x/dx f a C b 2 ˇ_ˇ ˇ ˇ ˇ ;ˇˇf0.a/ˇˇ; ˇ ˇf0.b/ˇˇ ! D 1 b a ˇ ˇ ˇ ˇ ˇ 1 b a Z b a f .x/dx f  a C b 2 ˇ_ˇ ˇ ˇ ˇ 1 .˛C 1/.˛ C 2/  1 1 2˛C1  ˇ ˇf0.a/ˇˇ  1 4 1 .˛_{C 1/.˛ C 2/}  1 1 2˛C1  ˇ ˇf0.b/ˇˇ 0; that is ˇ ˇ ˇ ˇ ˇ 1 b a Z b a f .x/dx f  a C b 2 ˇ_ˇ ˇ ˇ ˇ  .b a/  1 .˛_{C 1/.˛ C 2/}  1 1 2˛C1  ˇ ˇf0.a/ˇˇ C 1 4 1 .˛_{C 1/.˛ C 2/}  1 1 2˛C1  ˇ ˇf0.b/ ˇ ˇ 

which completes the proof. 

Corollary 3. If_jf0_{j is h convex on Œa; b ; then we have the inequalty} ˇ ˇ ˇ ˇ ˇ 1 b a Z b a f .x/dx f  a C b 2 ˇ_ˇ ˇ ˇ ˇ  .b a/ 0 @ 1 Z 0 h.t /_{jp.t/j dt} 1 A ˇ ˇf0.a/ˇˇC ˇ ˇf0.b/ˇˇ :

Proof. It is known that an h convex is an F convex. From (1.8) with w.t /D jp.t/j, we have TF ;w.u1; u2; u3/D u1 0 @ 1 Z 0 h.t /_{jp.t/j dt} 1 Au2 0 @ 1 Z 0 h.1 t /_{jp.t/j dt} 1 Au3 D u1 0 @ 1 Z 0 h.t /_{jp.t/j dt} 1 Au2 0 @ 1 Z 0 h.1 t /_{jp.1 t/j dt} 1 Au3 D u1 0 @ 1 Z 0 h.t /_{jp.t/j dt} 1 A.u2C u3/ for u1; u2; u32 R: Then, TF ;w 1 b a ˇ ˇ ˇ ˇ ˇ 1 b a Z b a f .x/dx f  a C b 2 ˇ_ˇ ˇ ˇ ˇ ;ˇˇf0.a/ˇˇ; ˇ ˇf0.b/ˇˇ ! D_b1_a ˇ ˇ ˇ ˇ ˇ 1 b a Z b a f .x/dx f a C b 2 ˇ_ˇ ˇ ˇ ˇ 0 @ 1 Z 0 h.t /_{jp.t/j dt} 1 A ˇ ˇf0.a/ˇˇC ˇ ˇf0.b/ˇˇ :

Thus, by Theorem4, we get 1 b a ˇ ˇ ˇ ˇ ˇ 1 b a Z b a f .x/dx f a C b 2 ˇ_ˇ ˇ ˇ ˇ 0 @ 1 Z 0 h.t /_{jp.t/j dt} 1 A ˇ ˇf0.a/ˇ ˇC ˇ ˇf0.b/ˇ ˇ  0

which completes the proof. 

Theorem 5. Letf _{W I}ı_{ R ! R be a differentiable mapping on I}ı; a; b_{2 I}ı; a < b and p > 1: Suppose that_jf0_jp=.p 1/isF convex on Œa; b ; for some F _{2 F} and_jf0_{j 2 L}p=.p 1/.a; b/: Then we have

TF ;1  A1.f; p/; ˇ ˇf0.a/ˇˇ p=.p 1/ ;ˇˇf0.b/ˇˇ p=.p 1/  0 where A1.f; p/D  2 b a _pp₁ .p_{C 1/}p11 ˇ ˇ ˇ ˇ ˇ 1 b a Z b a f .x/dx f  a C b 2 ˇ_ˇ ˇ ˇ ˇ p p 1 :

Proof. As_jf0_jp=.p 1/is F convex, we have Fˇˇf0.t a_{C .1 t/b/}ˇˇ p=.p 1/ ;ˇˇf0.a/ˇˇ p=.p 1/ ;ˇˇf0.b/ˇˇ p=.p 1/ ; t_{ 0; t 2 Œ0; 1 :} Using axiom (A2) with w.t /D 1, we get

TF ;1 0 @ 1 Z 0 ˇ ˇf0.t a_{C .1 t/b/}ˇˇ p=.p 1/ dt;ˇˇf0.a/ˇˇ p=.p 1/ ;ˇˇf0.b/ˇˇ p=.p 1/ 1 A 0:

Using Lemma1and H¨older inequality, we obtain ˇ ˇ ˇ ˇ ˇ 1 b a Z b a f .x/dx f a C b 2 ˇ_ˇ ˇ ˇ ˇ D .b a/ Z 1 0 jp .t/j ˇ ˇf0.t a_{C .1 t/b/}ˇˇdt  .b a/ Z 1 0 jp .t/j p_dt 1 pZ 1 0 ˇ ˇf0.t a_{C .1 t/b/}ˇˇ p=.p 1/ dt  p 1 p Db a 2  ₁ p_{C 1} 1_pZ 1 0 ˇ ˇf0.t a_{C .1 t/b/}ˇ ˇ p=.p 1/ dt  p 1 p ; that is A1.f; p/ Z 1 0 ˇ ˇf0.t a_{C .1 t/b/}ˇˇ p=.p 1/ dt:

Since TF ;1is nondecreasing with respect to the first variable, we can obtain the

de-sired result easily. 

Corollary 4. Ifjf0jp=.p 1/ is" convex on Œa; b ; "_{ 0; then we have} ˇ ˇ ˇ ˇ ˇ 1 b a Z b a f .x/dx f  a C b 2 ˇ_ˇ ˇ ˇ ˇ  b a 2.p_{C 1/}p1 " jf0.a/jp=.p 1/C jf0.b/jp=.p 1/ 2 C " #p_p1 : Proof. Using (1.3) with w.t /D 1; we have

TF ;1.u1; u2; u3/D u1

u2C u3

2 " (2.1)

for u1; u2; u32 R: Then, by the Theorem5, we have

0_TF ;1  A1.f; p/; ˇ ˇf0.a/ˇˇ p=.p 1/ ;ˇˇf0.b/ˇˇ p=.p 1/

DA1.f; p/ jf 0_.a/jp=.p 1/ C jf0.b/_jp=.p 1/ 2 " D  ₂ b a _pp₁ .pC 1/p11 ˇ ˇ ˇ ˇ ˇ 1 b a Z b a f .x/dx f  a C b 2 ˇ_ˇ ˇ ˇ ˇ p p 1 jf0.a/_jp=.p 1/_{C jf}0.b/_jp=.p 1/ 2 "; that is, ˇ ˇ ˇ ˇ ˇ 1 b a Z b a f .x/dx f  a C b 2 ˇ_ˇ ˇ ˇ ˇ  b a 2.p_{C 1/}p1 " jf0.a/_jp=.p 1/_{C jf}0.b/_jp=.p 1/ 2 C " #p_p1 :  Remark3. If we choose "D 0 in Corollary4, thenjf0jp=.p 1/ is convex and we have the inequality

ˇ ˇ ˇ ˇ ˇ 1 b a Z b a f .x/dx f  a C b 2 ˇ_ˇ ˇ ˇ ˇ  b a 2.p_{C 1/}p1 " jf0.a/_jp=.p 1/C jf0.b/_jp=.p 1/ 2 #p_p1 :

Corollary 5. If jf0jp=.p 1/ is ˛ convex, ˛2 .0; 1, then we have the following inequality ˇ ˇ ˇ ˇ ˇ 1 b a Z b a f .x/dx f  a C b 2 ˇ_ˇ ˇ ˇ ˇ  b a 2.p_{C 1/}p1 " jf0.a/_jp=.p 1/_{C ˛ jf}0.b/_jp=.p 1/ ˛_{C 1} #p_p1 : Proof. From (1.6) with w.t /D 1; we have

TF ;1.u1; u2; u3/D u1 0 @ 1 Z 0 t˛dt 1 Au2 0 @ 1 Z 0 .1 t˛/dt 1 Au3 (2.2) D u1 u2 ˛_{C 1} ˛ ˛_{C 1}u3

for u1; u2; u32 R: By the Theorem5, it follows that 0_TF ;1  A1.f; p/; ˇ ˇf0.a/ˇˇ p=.p 1/ ;ˇˇf0.b/ˇˇ p=.p 1/ DA1.f; p/ jf 0_.a/jp=.p 1/ ˛_{C 1} ˛ ˛_{C 1} ˇ ˇf0.b/ ˇ ˇ p=.p 1/ D  2 b a _pp₁ .p_{C 1/}p11 ˇ ˇ ˇ ˇ ˇ 1 b a Z b a f .x/dx f  a C b 2 ˇ_ˇ ˇ ˇ ˇ p p 1 jf0.a/_jp=.p 1/ ˛_{C 1} ˛ ˛_{C 1} ˇ ˇf0.b/ ˇ ˇ p=.p 1/ ;

which achieves the proof. 

Corollary 6. If_jf0_jp=.p 1/ ish convex, then we have the following inequality ˇ ˇ ˇ ˇ ˇ 1 b a Z b a f .x/dx f  a C b 2 ˇ_ˇ ˇ ˇ ˇ  b a 2.pC 1/p1 0 @ 1 Z 0 h.t /dt 1 A p 1 p h_ˇ ˇf0.a/ˇˇ p=.p 1/ Cˇˇf0.b/ˇˇ p=.p 1/ip_p1 :

Proof. Using (1.8) with w.t /D 1; we have

TF ;1.u1; u2; u3/Du1 0 @ 1 Z 0 h.t /dt 1 Au2 0 @ 1 Z 0 h.1 t /dt 1 Au3 (2.3) Du1 0 @ 1 Z 0 h.t /dt 1 A.u2C u3/

for u1; u2; u32 R: By the Theorem5, we obtain

0_TF ;1  A1.f; p/; ˇ ˇf0.a/ˇˇ p=.p 1/ ;ˇˇf0.b/ˇˇ p=.p 1/ DA1.f; p/ 0 @ 1 Z 0 h.t /dt 1 A _ˇ ˇf0.a/ ˇ ˇ p=.p 1/ Cˇ ˇf0.b/ ˇ ˇ p=.p 1/ D  2 b a _pp₁ .p_{C 1/}p11 ˇ ˇ ˇ ˇ ˇ 1 b a Z b a f .x/dx f  a C b 2 ˇ_ˇ ˇ ˇ ˇ p p 1

0 @ 1 Z 0 h.t /dt 1 A _ˇ ˇf0.a/ˇˇ p=.p 1/ Cˇˇf0.b/ˇˇ p=.p 1/ :

This completes the proof. 

3. TRAPEZOID TYPE INEQUALITIES FOR TWICE DIFFERENTIABLE FUNCTIONS

In this section, we obtain some trapezoid type inequalities for functions whose second derivatives absolute values are F convex.

Theorem 6. LetI _{ R be an interval, f W I}ı_{ R ! R be a twice differentiable} mapping onIı; .a; b/2 Iı Iı; a < b: Suppose thatjf00j is F convex on Œa; b ; for someF _{2 F and the function t 2 Œ0; 1 ! L}w.t /belongs toL1Œ0; 1 ; where w.t /D

t .1 t /: Then we have the following inequality TF ;w 2 .b a/2 ˇ ˇ ˇ ˇ ˇ f .a/_{C f .b/} 2 1 b a Z b a f .x/dx ˇ ˇ ˇ ˇ ˇ ;ˇˇf00.a/ˇˇ; ˇ ˇf00.b/ˇˇ ! C 1 Z 0 Lw.t /dt 0:

Proof. Since_jf00_{j is F convex, we have} F ˇˇf00.t a_{C .1 t/b/}ˇˇ;

ˇ

ˇf00.a/ˇˇ; ˇ

ˇf00.b/ˇˇ; t
 0; t 2 Œ0;1: Multiplying this inequality by w.t / and using axiom (A3), we get F w.t /ˇˇf00.t a_{C .1 t/b/}ˇˇ; w.t /

ˇ

ˇf00.a/ˇˇ; w.t / ˇ

ˇf00.b/ˇˇ; t
C L_{w.t /} 0; t 2 Œ0; 1 : Integrating over .0; 1/ with respect to the variable t and using axiom (A2), we obtain TF ;w 0 @ 1 Z 0 w.t /ˇˇf00.t a_{C .1 t/b/}ˇˇdt; ˇ ˇf00.a/ˇˇ; ˇ ˇf00.b/ˇˇ 1 AC 1 Z 0 Lw.t /dt 0; t 2 Œ0; 1 :

On the other hand using Lemma2, we have 2 .b a/2 ˇ ˇ ˇ ˇ ˇ f .a/_{C f .b/} 2 1 b a Z b a f .x/dx ˇ ˇ ˇ ˇ ˇ  Z 1 0 t .1 t /ˇˇf00.t a_{C .1 t/b/}ˇˇdt: Since TF ;w is nondecreasing with respect to the first variable, we get

TF ;w 2 .b a/2 ˇ ˇ ˇ ˇ ˇ f .a/_{C f .b/} 2 1 b a Z b a f .x/dx ˇ ˇ ˇ ˇ ˇ ;ˇˇf00.a/ˇˇ; ˇ ˇf00.b/ˇˇ ! C 1 Z 0 Lw.t /dt 0:

This completes the proof.  Corollary 7. Letf _{W I}ı_{ R ! R be twice differentiable function on I}ı,a; b_{2 I}ı witha < b: Suppose that the function_jf00_{j is " convex on Œa; b ; "  0: Then we have}

ˇ ˇ ˇ ˇ ˇ f .a/_{C f .b/} 2 1 b a Z b a f .x/dx ˇ ˇ ˇ ˇ ˇ  .b a/ 2 2  jf00.a/_{j C jf}00.b/_j 12 C " 6  : Proof. We know that an " convex is an F convex. Using (1.4) with w.t /D t .1 t /; we obtain 1 Z 0 Lw.t /dt D " 1 Z 0 .1 w.t //dt _D5" 6 : From (1.3) with w.t /D t.1 t/; we get

TF ;w.u1; u2; u3/Du1 0 @ 1 Z 0 t2.1 t /dt 1 Au2 0 @ 1 Z 0 t .1 t /2dt 1 Au3 " Du1 u2C u3 12 " for u1; u2; u32 R: Hence, TF ;w 2 .b a/2 ˇ ˇ ˇ ˇ ˇ f .a/_{C f .b/} 2 1 b a Z b a f .x/dx ˇ ˇ ˇ ˇ ˇ ;ˇˇf00.a/ˇˇ; ˇ ˇf00.b/ˇˇ ! D 2 .b a/2 ˇ ˇ ˇ ˇ ˇ f .a/_{C f .b/} 2 1 b a Z b a f .x/dx ˇ ˇ ˇ ˇ ˇ jf00.a/_{j C jf}00.b/_j 12 ":

Thus, by Theorem6, we have 2 .b a/2 ˇ ˇ ˇ ˇ ˇ f .a/_{C f .b/} 2 1 b a Z b a f .x/dx ˇ ˇ ˇ ˇ ˇ jf00.a/_{j C jf}00.b/_j 12 "C 5" 6  0:

This completes the proof. 

Remark4. If we taking "D 0 in Corollary7, thenjf00j is convex and we have the inequality ˇ ˇ ˇ ˇ ˇ f .a/_{C f .b/} 2 1 b a Z b a f .x/dx ˇ ˇ ˇ ˇ ˇ  .b a/ 2 12  jf00.a/_{j C jf}00.b/_j 2  : Corollary 8. Letf _{W I}ı_{ R ! R be twice differentiable function on I}ı,a; b_{2 I}ı witha < b: Suppose that the functionjf00j is ˛ convex on Œa; b ; ˛ 2 .0; 1: Then we

have ˇ ˇ ˇ ˇ ˇ f .a/_{C f .b/} 2 1 b a Z b a f .x/dx ˇ ˇ ˇ ˇ ˇ .b a/ 2 2  jf00.a/_j .˛_{C 2/.˛ C 3/}  1 6 1 .˛_{C 2/.˛ C 3/}  ˇ ˇf00.b/ˇˇ  :

Proof. We know that an ˛ convex is an F convex. From (1.6) with w.t /D t .1 t /, we have TF ;w.u1; u2; u3/Du1 0 @ 1 Z 0 t˛C1.1 t /dt 1 Au2 0 @ 1 Z 0 .1 t˛/t .1 t /dt 1 Au3 Du1 u2 .˛_{C 2/.˛ C 3/}  1 6 1 .˛_{C 2/.˛ C 3/}  u3

for u1; u2; u32 R: It follows that

TF ;w 2 .b a/2 ˇ ˇ ˇ ˇ ˇ f .a/_{C f .b/} 2 1 b a Z b a f .x/dx ˇ ˇ ˇ ˇ ˇ ;ˇˇf00.a/ˇˇ; ˇ ˇf00.b/ˇˇ ! D 2 .b a/2 ˇ ˇ ˇ ˇ ˇ f .a/_{C f .b/} 2 1 b a Z b a f .x/dx ˇ ˇ ˇ ˇ ˇ jf00.a/j .˛C 2/.˛ C 3/  1 6 1 .˛C 2/.˛ C 3/  ˇ ˇf00.b/ˇˇ: Consequently, by Theorem6, we obtain

2 .b a/2 ˇ ˇ ˇ ˇ ˇ f .a/C f .b/ 2 1 b a Z b a f .x/dx ˇ ˇ ˇ ˇ ˇ jf00.a/_j .˛C 2/.˛ C 3/  1 6 1 .˛C 2/.˛ C 3/  ˇ ˇf00.b/ˇˇ 0;

which completes the proof. 

Corollary 9. Letf _{W I}ı_{ R ! R be twice differentiable function on I}ı,a; b_{2 I}ı witha < b: Suppose that the function_jf00_{j is h convex on Œa; b : Then we have}

ˇ ˇ ˇ ˇ ˇ f .a/_{C f .b/} 2 1 b a Z b a f .x/dx ˇ ˇ ˇ ˇ ˇ  .b a/2 0 @ 1 Z 0 h.t /t .1 t /dt 1 A  jf00.a/_{j C jf}00.b/_j 2  :

Proof. We know that an h convex is an F convex. From (1.8) with w.t /D t .1 t /, we have TF ;w.u1; u2; u3/Du1 0 @ 1 Z 0 h.t /w.t /dt 1 Au2 0 @ 1 Z 0 h.1 t /w.t /dt 1 Au3 Du1 0 @ 1 Z 0 h.t /t .1 t /dt 1 Au2 0 @ 1 Z 0 h.1 t /t .1 t /dt 1 Au3 Du1 0 @ 1 Z 0 h.t /t .1 t /dt 1 A.u2C u3/ for u1; u2; u32 R: Then, TF ;w 2 .b a/2 ˇ ˇ ˇ ˇ ˇ f .a/_{C f .b/} 2 1 b a Z b a f .x/dx ˇ ˇ ˇ ˇ ˇ ;ˇ ˇf00.a/ˇ ˇ; ˇ ˇf00.b/ˇ ˇ ! D 2 .b a/2 ˇ ˇ ˇ ˇ ˇ f .a/_{C f .b/} 2 1 b a Z b a f .x/dx ˇ ˇ ˇ ˇ ˇ 0 @ 1 Z 0 h.t /t .1 t /dt 1 A ˇ ˇf00.a/ˇˇC ˇ ˇf00.b/ˇˇ :

Thus, by Theorem6, we get

D 2 .b a/2 ˇ ˇ ˇ ˇ ˇ f .a/_{C f .b/} 2 1 b a Z b a f .x/dx ˇ ˇ ˇ ˇ ˇ 0 @ 1 Z 0 h.t /t .1 t /dt 1 A ˇ ˇf00.a/ˇˇC ˇ ˇf00.b/ˇˇ  0:

This finishes the proof. 

Theorem 7. Letf _{W I}ı_{ R ! R be a twice differentiable mapping on I}ı; a; b₂ Iı; a < b and let p > 1: Suppose that_jf00_jp=.p 1/ isF convex on Œa; b ; for some F 2 F and jf00j 2 Lp=.p 1/.a; b/: Then we have

TF ;1  A2.f; p/; ˇ ˇf00.a/ˇˇ p=.p 1/ ;ˇˇf00.b/ˇˇ p=.p 1/  0 where A2.f; p/D

 2 .b a/2 _pp₁ 1 B.pC 1; p C 1/ _p1₁ˇ_ˇ ˇ ˇ ˇ f .a/C f .b/ 2 1 b a Z b a f .x/dx ˇ ˇ ˇ ˇ ˇ p p 1

andB.x; y/ is Euler beta function defined by B.x; y/_D

Z 1

0

tx 1.1 t /y 1dt: Proof. Sincejf00jp=.p 1/is F convex, we have Fˇˇf00.t aC .1 t/b/ˇ ˇ p=.p 1/ ;ˇˇf00.a/ˇˇ p=.p 1/ ;ˇˇf00.b/ˇˇ p=.p 1/ ; t 0; t 2 Œ0; 1 : Using axiom (A2) with w.t /D 1, we get

TF ;1 0 @ 1 Z 0 ˇ ˇf00.t aC .1 t/b/ˇ ˇ p=.p 1/ dt;ˇˇf00.a/ˇˇ p=.p 1/ ;ˇˇf00.b/ˇˇ p=.p 1/ 1 A 0:

Using Lemma2and H¨older inequality, we obtain ˇ ˇ ˇ ˇ ˇ f .a/_{C f .b/} 2 1 b a Z b a f .x/dx ˇ ˇ ˇ ˇ ˇ D.b a/ 2 2 Z 1 0 jt.1 t/j ˇ ˇf00.t aC .1 t/b/ˇ ˇdt .b a/ 2 2 Z 1 0 tp.1 t /pdt  1 pZ 1 0 ˇ ˇf00.t a_{C .1 t/b/}ˇˇ p=.p 1/ dt  p 1 p D.b a/ 2 2 .B.pC 1; p C 1// 1 p Z 1 0 ˇ ˇf00.t a_{C .1 t/b/}ˇˇ p=.p 1/ dt  p 1 p ; that is A2.f; p/ Z 1 0 ˇ ˇf00.t a_{C .1 t/b/}ˇˇ p=.p 1/ dt: Since TF ;1is nondecreasing with respect to the first variable, we have

TF ;1  A2.f; p/; ˇ ˇf00.a/ˇˇ p=.p 1/ ;ˇˇf00.b/ˇˇ p=.p 1/  0:  Corollary 10. If_jf00_jp=.p 1/is" convex, we have the following inequality

ˇ ˇ ˇ ˇ ˇ f .a/C f .b/ 2 1 b a Z b a f .x/dx ˇ ˇ ˇ ˇ ˇ

.b a/ 2 2 .B.pC 1; p C 1// 1 p " jf00.a/_jp=.p 1/_{C jf}00.b/_jp=.p 1/ 2 C " #p_p1 : Proof. Using (2.1) and by Theorem7, it can be proved easily. It is omitted.  Remark5. If we choose "D 0 in Corollary10, then we have

ˇ ˇ ˇ ˇ ˇ f .a/_{C f .b/} 2 1 b a Z b a f .x/dx ˇ ˇ ˇ ˇ ˇ .b a/ 2 2 .B.pC 1; p C 1// 1 p " jf00.a/_jp=.p 1/_{C jf}00.b/_jp=.p 1/ 2 #p_p1 :

Corollary 11. If_jf00_jp=.p 1/is˛ convex, we have the following inequality ˇ ˇ ˇ ˇ ˇ f .a/_{C f .b/} 2 1 b a Z b a f .x/dx ˇ ˇ ˇ ˇ ˇ .b a/ 2 2 .B.pC 1; p C 1// 1 p " jf0.a/_jp=.p 1/_{C ˛ jf}0.b/_jp=.p 1/ ˛_{C 1} #p_p1 : Proof. Using (2.2) and by Theorem7, it can be proved easily. It is omitted.  Corollary 12. Ifjf00jp=.p 1/ish convex, we have the following inequality

ˇ ˇ ˇ ˇ ˇ f .a/C f .b/ 2 1 b a Z b a f .x/dx ˇ ˇ ˇ ˇ ˇ  .b a/ 2 2 .B.pC 1; p C 1// 1 p 0 @ 1 Z 0 h.t /dt 1 A p 1 p h_ˇ ˇf0.a/ ˇ ˇ p=.p 1/ Cˇ ˇf0.b/ ˇ ˇ p=.p 1/ip_p1 :

Proof. Using (2.3) and by Theorem7, it can be proved easily. It is omitted.  4. MIDPOINT TYPE INEQUALITIES FOR TWICE DIFFERENTIABLE FUNCTIONS

In this section, we prove some midpoint type inequalities for functions whose second derivatives absolute values are F convex.

Theorem 8. LetI  R be an interval, f W Iı R ! R be a twice differentiable mapping onIı; .a; b/2 Iı Iı; a < b: Suppose thatjf00j is F convex on Œa; b ;

for someF 2 F the function t 2 Œ0; 1 ! Lm.t /belongs toL1Œ0; 1 : If F is linear

with respect to the first, second and third variables, then TF ;m  4 .b a/2 ˇ ˇ ˇ ˇ 1 b a Rb af .x/dx f  a C b 2 ˇ ˇ ˇ ˇ ;jf00_{.a/j C jf}00_{.b/j ; jf}00_{.a/j C jf}00_.b/j  C 1 R 0 Lm.t /dt 0 where m.t /_D 8 < : t2; t₂0;1 2  .1 t /2; t₂ 1₂; 1 : Proof. Sincejf00j is F convex, we have

F ˇˇf00.t aC .1 t/b/ˇ ˇ; ˇ ˇf00.a/ˇˇ; ˇ ˇf00.b/ˇˇ; t
 0; t 2 Œ0;1 and F ˇˇf00.t b_{C .1 t/a/}ˇˇ; ˇ ˇf00.b/ˇˇ; ˇ ˇf00.a/ˇˇ; t
 0; t 2 Œ0;1: Using the linearity of F , we obtain

F ˇˇf00.t a_{C .1 t/b/}ˇˇC ˇ ˇf00.t b_{C .1 t/a/}ˇˇ; ˇ ˇf00.b/ˇˇC ˇ ˇf00.a/ˇˇ; ˇ ˇf00.b/ˇˇC ˇ ˇf00.a/ˇˇ; t
 0;

for t2 Œ0; 1 : Multiplying this inequality by m.t/ and using axiom (A3), we get F m.t /ˇ ˇf00.t a_{C .1 t/b/}ˇˇC ˇ ˇf00.t b_{C .1 t/a/}ˇˇ ; m.t/ ˇ ˇf00.b/ˇˇC ˇ ˇf00.a/ˇˇ ; m.t /ˇ ˇf00.b/ˇˇC ˇ ˇf00.a/ˇˇ ; t
C L_{m.t /} 0;

for t 2 Œ0; 1 : Integrating over Œ0; 1 with respect to the variable t and using axiom (A2), we obtain TF ;m 1 Z 0 m.t /ˇ ˇf00.t a_{C .1 t/b/}ˇˇC ˇ ˇf00.t b_{C .1 t/a/}ˇˇ dt; ˇ ˇf00.b/ˇˇC ˇ ˇf00.a/ˇˇ; ˇ ˇf00.b/ˇˇC ˇ ˇf00.a/ˇˇ ! C 1 Z 0 Lm.t /dt  0;

for t2 Œ0; 1 : On the other hand, using Lemma3, we have 4 .b a/2 ˇ ˇ ˇ ˇ ˇ 1 b a Z b a f .x/dx f a C b 2 ˇ_ˇ ˇ ˇ ˇ  Z 1 0 m .t /ˇ ˇf00.t a_{C .1 t/b/}ˇˇC ˇ ˇf00.t b_{C .1 t/a/}ˇˇ dt:

Since TF ;mis nondecreasing with respect to the first variable, we get TF ;m 4 .b a/2 ˇ ˇ ˇ ˇ ˇ 1 b a Z b a f .x/dx f  a C b 2 ˇ_ˇ ˇ ˇ ˇ ; ˇ ˇf00.a/ˇˇC ˇ ˇf00.b/ˇˇ; ˇ ˇf00.a/ˇˇC ˇ ˇf00.b/ˇˇ ! C 1 Z 0 Lm.t /dt 0:

This completes the proof. 

Corollary 13. Letf _{W I}ı_{ R ! R be twice differentiable function on I}ı,a; b_{2 I}ı witha < b: Suppose that the function_jf00_{j is " convex on Œa; b ; "  0: Then we have}

ˇ ˇ ˇ ˇ ˇ 1 b a Z b a f .x/dx f a C b 2 ˇ_ˇ ˇ ˇ ˇ  .b a/ 2 4  jf00.a/_{j C jf}00.b/_j 12 C " 12  : Proof. Using (1.4) with w.t /D m.t/; we obtain

1 Z 0 Lm.t /dtD" 1 Z 0 .1 m.t //dt D" 0 B @ 1=2 Z 0 .1 t2/dt_C 1 Z 1=2 .1 .1 t /2/dt 1 C A D11 12": From (1.3) with w.t /D m.t/; we get

TF ;w.u1; u2; u3/Du1 0 @ 1 Z 0 t m.t /dt 1 Au2 0 @ 1 Z 0 .1 t /m.t /dt 1 Au3 " Du1 u2C u3 24 " for u1; u2; u32 R: Hence, TF ;m 4 .b a/2 ˇ ˇ ˇ ˇ ˇ 1 b a Z b a f .x/dx f  a C b 2 ˇ_ˇ ˇ ˇ ˇ ; ˇ ˇf00.b/ˇˇC ˇ ˇf00.a/ˇˇ; ˇ ˇf00.b/ˇˇC ˇ ˇf00.a/ˇˇ ! D 4 .b a/2 ˇ ˇ ˇ ˇ ˇ 1 b a Z b a f .x/dx f  a C b 2 ˇ_ˇ ˇ ˇ ˇ jf00.a/_{j C jf}00.b/_j 12 ":

Thus, by Theorem8, we have 4 .b a/2 ˇ ˇ ˇ ˇ ˇ 1 b a Z b a f .x/dx f  a C b 2 ˇ_ˇ ˇ ˇ ˇ jf00.a/_{j C jf}00.b/_j 12 "C 11 12" 0; that is ˇ ˇ ˇ ˇ ˇ 1 b a Z b a f .x/dx f a C b 2 ˇ_ˇ ˇ ˇ ˇ  .b a/ 2 4  jf00.a/_{j C jf}00.b/_j 12 C " 12  :

This completes the proof. 

Remark6. If we taking "D 0 in Corollary13, thenjf00j is convex and we have the inequality ˇ ˇ ˇ ˇ ˇ 1 b a Z b a f .x/dx f  a C b 2 ˇ_ˇ ˇ ˇ ˇ .b a/ 2 24  jf00.a/_{j C jf}00.b/_j 2 

which is given by Sarikaya et al. in [13].

Corollary 14. Letf _{W I}ı_{ R ! R be twice differentiable function on I}ı,a; b_{2 I}ı witha < b: Suppose that the function_jf00_{j is ˛ convex on Œa; b ; ˛ 2 .0; 1: Then we} have ˇ ˇ ˇ ˇ ˇ 1 b a Z b a f .x/dx f  a C b 2 ˇ_ˇ ˇ ˇ ˇ  .b a/ 2 24  jf00.a/_{j C jf}00.b/_j 2  : Proof. From (1.6) with w.t /D m.t/, we have

TF ;w.u1; u2; u3/D u1 0 @ 1 Z 0 t˛m.t /dt 1 Au2 0 @ 1 Z 0 .1 t˛/m.t /dt 1 Au3 D u1 1 .˛C 1/ .˛ C 2/  2 ˛C 3 1 2˛C1  u2  1 12 1 .˛_{C 1/ .˛ C 2/}  ₂ ˛_{C 3} 1 2˛C1  u3

for u1; u2; u32 R: It follows that

TF ;w 4 .b a/2 ˇ ˇ ˇ ˇ ˇ 1 b a Z b a f .x/dx f  a C b 2 ˇ_ˇ ˇ ˇ ˇ ; ˇ ˇf00.b/ˇˇC ˇ ˇf00.a/ˇˇ; ˇ ˇf00.b/ˇˇC ˇ ˇf00.a/ˇˇ ! D 4 .b a/2 ˇ ˇ ˇ ˇ ˇ 1 b a Z b a f .x/dx f  a C b 2 ˇ_ˇ ˇ ˇ ˇ

1 .˛_{C 1/ .˛ C 2/}  2 ˛_{C 3} 1 2˛C1  ˇ ˇf00.b/ˇˇC ˇ ˇf00.a/ˇˇ   1 12 1 .˛_{C 1/ .˛ C 2/}  ₂ ˛_{C 3} 1 2˛C1  ˇ ˇf00.b/ˇˇC ˇ ˇf00.a/ˇˇ  D 4 .b a/2 ˇ ˇ ˇ ˇ ˇ 1 b a Z b a f .x/dx f  a C b 2 ˇ_ˇ ˇ ˇ ˇ jf00.b/_{j C jf}00.a/_j 12 :

which completes the proof. 

Corollary 15. Letf _{W I}ı_{ R ! R be twice differentiable function on I}ı,a; b_{2 I}ı witha < b: Suppose that the function_jf00_{j is h convex on Œa; b : Then we have}

ˇ ˇ ˇ ˇ ˇ f .a/C f .b/ 2 1 b a Z b a f .x/dx ˇ ˇ ˇ ˇ ˇ  .b a/2 0 @ 1 Z 0 h.t /t .1 t /dt 1 A  jf00.a/_{j C jf}00.b/_j 2  :

Proof. From (1.8) with w.t /D m.t/, we have

TF ;w.u1; u2; u3/D u1 0 @ 1 Z 0 h.t /m.t /dt 1 Au2 0 @ 1 Z 0 h.1 t /m.t /dt 1 Au3 D u1 0 @ 1 Z 0 h.t /m.t /dt 1 Au2 0 @ 1 Z 0 h.t /m.1 t /dt 1 Au3 D u1 0 @ 1 Z 0 h.t /m.t /dt 1 A.u2C u3/

for u1; u2; u32 R: Then, by Theorem8,

TF ;w 4 .b a/2 ˇ ˇ ˇ ˇ ˇ 1 b a Z b a f .x/dx f  a C b 2 ˇ_ˇ ˇ ˇ ˇ ; ˇ ˇf00.b/ˇˇC ˇ ˇf00.a/ˇˇ; ˇ ˇf00.b/ˇˇC ˇ ˇf00.a/ˇˇ ! D 4 .b a/2 ˇ ˇ ˇ ˇ ˇ 1 b a Z b a f .x/dx f  a C b 2 ˇ_ˇ ˇ ˇ ˇ

2 0 @ 1 Z 0 h.t /m.t /dt 1 A ˇ ˇf00.a/ˇˇC ˇ ˇf00.b/ˇˇ  0 that is, ˇ ˇ ˇ ˇ ˇ 1 b a Z b a f .x/dx f  a C b 2 ˇ_ˇ ˇ ˇ ˇ  .b a/2 0 @ 1 Z 0 h.t /m.t /dt 1 A  jf00.a/_{j C jf}00.b/_j 2 

which completes the proof. 

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Authors’ addresses

H. Budak

D¨uzce University, Faculty of Science and Arts, Department of Mathematics D¨uzce, Turkey E-mail address: hsyn.budak@gmail.com

T. Tunc¸

D¨uzce University, Faculty of Science and Arts, Department of Mathematics D¨uzce, Turkey E-mail address: tubatunc03@gmail.com

M. Z. Sarikaya

D¨uzce University, Faculty of Science and Arts, Department of Mathematics D¨uzce, Turkey E-mail address: sarikayamz@gmail.com