Some ostrowski's type inequalities for functions whose second derivatives are s-convex in the second sense

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Vol. XLVII No 1 2014

Erhan Set, Mehmet Zeki Sarikaya, M. Emin Ozdemir

SOME OSTROWSKI’S TYPE INEQUALITIES FOR FUNCTIONS WHOSE SECOND DERIVATIVES

ARE s-CONVEX IN THE SECOND SENSE

Abstract. Some new inequalities of the Ostrowski type for twice differentiable mappings whose derivatives in absolute value are s-convex in the second sense are given.

1. Introduction

In 1938, Ostrowski proved the following integral inequality [12]:

Theorem 1. Let I Ď R, f : I Ñ R be a differentiable mapping on pa, bq whose derivative f1 : pa, bq Ñ R is bounded on pa, bq, i.e., }f1_}

8 “

sup tPpa,bq

|f1ptq| ă 8. Then, the inequality holds: ˇ ˇ ˇ ˇf pxq ´ 1 b ´ a żb a f ptqdt ˇ ˇ ˇ ˇ ≤ « 1 4` `x ´ a`b 2 ˘2 pb ´ aq2 ff pb ´ aq››f1 › › 8,

for all x P ra, bs . The constant 1₄ is sharp in the sense that it cannot be replaced by a smaller one.

For some applications of Ostrowski’s inequality see ([1]–[4]) and for recent results and generalizations concerning Ostrowski’s inequality see ([1]–[8]).

The class of s-convexity in the second sense is defined in the following way [9, 11]: a function f : r0, 8q Ñ R is said to be s-convex in the second sense if

f ptx ` p1 ´ tqyq ≤ tsf pxq ` p1 ´ tqsf pyq,

for all x, y P r0, 8q, t P r0, 1s and some fixed s P p0, 1s. This class is usually denoted by K_s2.

In [10], Dragomir and Fitzpatrick proved the Hadamard inequality for s-convex functions in the second sense:

2010 Mathematics Subject Classification: 26A15, 26D07, 26D15, 26D10.

Key words and phrases: Ostrowski’s inequality, convex function, s-convex function.

DOI: 10.2478/dema-2014-0003

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Theorem 2. Suppose that f : r0, 8q Ñ r0, 8q is an s-convex function in the second sense, where s P p0, 1q, and let a, b P r0, 8q, a ă b. If f P L1pra, bsq then the following inequalities hold:

(1.1) 2s´1fˆ a ` b 2 ˙ ≤ 1 b ´ a b ż a f pxqdx ≤ f paq ` f pbq s ` 1 .

The constant k “ _s`11 is the best possible in the second inequality in (1.1).

In [3], Cerone et al. proved the following inequalities of Ostrowski type and Hadamard type, respectively.

Theorem 3. Let f : ra, bs Ñ R be a twice differentiable mapping on pa, bq and f2 _{: pa, bq Ñ R is bounded, i.e. }f}2

}₈ “ sup tPpa,bq

|f2ptq| ă 8. Then we have the inequality:

(1.2) ˇ ˇ ˇ ˇf pxq ´ 1 b ´ a żb a f ptqdt ´ ˆ x ´a ` b 2 ˙ f1pxq ˇ ˇ ˇ ˇ ≤ « 1 24pb ´ aq 2 ` 1 2 ˆ x ´a ` b 2 ˙2ff › ›f2 › › 8 ≤ pb ´ aq2 6 › ›f2 › › 8, for all x P ra, bs.

Corollary 1. Under the above assumptions, we have the mid-point in-equality: (1.3) ˇ ˇ ˇ ˇf p a ` b 2 q ´ 1 b ´ a żb a f pxqdx ˇ ˇ ˇ ˇ ≤ pb ´ aq 2 24 › ›f2 › › 8.

In this article, we establish new Ostrowski’s type inequalities for s-convex functions in the second sense.

2. Main results

In order to establish our main results we need the following lemma. Lemma 1. Let I Ď R, f : I Ñ R be a twice differentiable function on I˝ with f2

P L1ra, bs where a, b P I with a ă b. Then

(2.1) 1 b ´ a żb a f puqdu ´ f pxq ` ˆ x ´a ` b 2 ˙ f1pxq “ px ´ aq 3 2pb ´ aq ż1 0 t2f2 ptx ` p1 ´ tqaqdt `pb ´ xq 3 2pb ´ aq ż1 0 t2f2 ptx ` p1 ´ tqbqdt, for each x P ra, bs.

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Proof. By integration by parts, we have the following identity (2.2) ż1 0 t2f2ptx ` p1 ´ tqaqdt “ t 2 px ´ aqf 1 ptx ` p1 ´ tqaq ˇ ˇ ˇ ˇ 1 0 ´ 2 x ´ a ż1 0 tf1 ptx ` p1 ´ tqaqdt “ f 1 pxq px ´ aq´ 2 x ´ a « t px ´ aqf ptx ` p1 ´ tqaq ˇ ˇ ˇ ˇ 1 0 ´ 1 x ´ a ż1 0 f ptx ` p1 ´ tqaqdt  “ f 1 pxq px ´ aq´ 2f pxq px ´ aq2` 2 px ´ aq2 ż1 0 f ptx ` p1 ´ tqaqdt.

By using the change of the variable u “ tx ` p1 ´ tqa for t P r0, 1s and multiplying the both sides of (2.2) by px´aq_2pb´aq3, we obtain

(2.3) px ´ aq 3 2pb ´ aq ż1 0 t2f2ptx ` p1 ´ tqaqdt “ px ´ aq 2_f1 pxq 2pb ´ aq ´ px ´ aqf pxq b ´ a ` 1 b ´ a żx a f puqdu. Similarly, we observe that

(2.4) pb ´ xq 3 2pb ´ aq ż1 0 t2f2ptx ` p1 ´ tqbqdt “ ´pb ´ xq 2_f1 pxq 2pb ´ aq ´ pb ´ xqf pxq b ´ a ` 1 b ´ a żb x f puqdu. Thus, adding (2.3) and (2.4) we get the required identity (2.1).

The following result may be stated:

Theorem 4. Let I Ă r0, 8q, f : I Ñ R be a twice differentiable function on I˝ _{such that f}2

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Proof. By Lemma 1 and by s-convex of |f2 |, we have ˇ ˇ ˇ ˇ 1 b ´ a żb a f puqdu ´ f pxq ` ˆ x ´a ` b 2 ˙ f1 pxq ˇ ˇ ˇ ˇ ≤ px ´ aq 3 2pb ´ aq ż1 0 t2ˇˇf2ptx ` p1 ´ tqaq ˇ ˇdt ` pb ´ xq 3 2pb ´ aq ż1 0 t2ˇˇf2ptx ` p1 ´ tqbq ˇ ˇdt ≤ px ´ aq 3 2pb ´ aq ż1 0 t2“tsˇˇf2pxq ˇ ˇ` p1 ´ tqs ˇ ˇf2paq ˇ ˇ‰ dt ` pb ´ xq 3 2pb ´ aq ż1 0 t2“tsˇˇf2pxq ˇ ˇ` p1 ´ tqs ˇ ˇf2pbq ˇ ˇ‰ dt “ px ´ aq 3 2pb ´ aq ż1 0 `ts`2ˇ ˇf2pxq ˇ ˇ` t2p1 ´ tqs ˇ ˇf2paq ˇ ˇ˘ dt ` pb ´ xq 3 2pb ´ aq ż1 0 `ts`2ˇ ˇf2pxq ˇ ˇ` t2p1 ´ tqs ˇ ˇf2pbq ˇ ˇ˘ dt “ px ´ aq 3 2pb ´ aq „ |f2pxq| s ` 3 ` 2 |f2 paq| ps ` 1qps ` 2qps ` 3q  ` pb ´ xq 3 2pb ´ aq „ |f2pxq| s ` 3 ` 2 |f2 pbq| ps ` 1qps ` 2qps ` 3q  “ 1 2pb ´ aq "„ |f2pxq| s ` 3 ` 2 |f2 paq| ps ` 1qps ` 2qps ` 3q  px ´ aq3 ` „ |f2pxq| s ` 3 ` 2 |f2 pbq| ps ` 1qps ` 2qps ` 3q  pb ´ xq3 * , where we have used the fact that

ż1 0 ts`2dt “ 1 s ` 3 and ż1 0 t2p1 ´ tqsdt “ 2 ps ` 1qps ` 2qps ` 3q. This completes the proof.

Corollary 2. If we put M “ sup xPra,bs

|f2| in Theorem 4, then we get

(2.6) ˇ ˇ ˇ ˇ 1 b ´ a żb a f puqdu ´ f pxq ` ˆ x ´a ` b 2 ˙ f1pxq ˇ ˇ ˇ ˇ ≤ 3M ˆ s2` 3s ` 4 ps ` 1qps ` 2qps ` 3q ˙« 1 24pb ´ aq 2 `1 2 ˆ x ´ a ` b 2 ˙2ff ≤ Mpb ´ aq 2 2 ˆ s2_{` 3s ` 4} ps ` 1qps ` 2qps ` 3q ˙ .

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Here, simple computation shows that px ´ aq3` pb ´ xq3 “ pb ´ aq « pb ´ aq2 4 ` 3 ˆ x ´a ` b 2 ˙2ff .

Remark 1. If in Corollary 2 we choose s “ 1, then we recapture the inequality (1.2) for functions f with convex |f2

| .

Corollary 3. If in Corollary 2 we choose x “ a`b₂ , then we get the mid-point inequality ˇ ˇ ˇ ˇ 1 b ´ a żb a f puqdu ´ f pa ` b 2 q ˇ ˇ ˇ ˇ ≤ Mpb ´ aq 2 2 ˆ s2` 3s ` 4 ps ` 1qps ` 2qps ` 3q ˙ .

Theorem 5. Let I Ă r0, 8q, f : I Ñ R be a twice differentiable function on I˝ _{such that f}2

P L1ra, bs where a, b P I with a ă b. If |f2|q is s-convex in the second sense on ra, bs, for some fixed s P p0, 1s, p, q ą 1 and 1_p ` 1_q “ 1, then the following inequality holds:

(2.7) ˇ ˇ ˇ ˇ 1 b ´ a żb a f puqdu ´ f pxq ` ˆ x ´a ` b 2 ˙ f1 pxq ˇ ˇ ˇ ˇ ≤ px ´ aq 3 2pb ´ aq ˆ 1 2p ` 1 ˙1 pˆ_|f2_pxq|q_{` |f}2_paq|q s ` 1 ˙1_q `pb ´ xq 3 2pb ´ aq ˆ 1 2p ` 1 ˙1 pˆ_|f2_pxq|q_{` |f}2_pbq|q s ` 1 ˙1 q , for each x P ra, bs.

Proof. Suppose that p ą 1. From Lemma 1 and by the Hölder inequality, we have ˇ ˇ ˇ ˇ 1 b ´ a żb a f puqdu ´ f pxq ` ˆ x ´a ` b 2 ˙ f1pxq ˇ ˇ ˇ ˇ ≤ px ´ aq 3 2pb ´ aq ż1 0 t2ˇˇf2ptx ` p1 ´ tqaq ˇ ˇdt ` pb ´ xq3 2pb ´ aq ż1 0 t2ˇˇf2ptx ` p1 ´ tqbq ˇ ˇdt ≤ px ´ aq 3 2pb ´ aq ˆż1 0 t2pdt ˙1_pˆż1 0 ˇ ˇf2ptx ` p1 ´ tqaq ˇ ˇ q dt ˙1_q `pb ´ xq 3 2pb ´ aq ˆż1 0 t2pdt ˙1_pˆż1 0 ˇ ˇf2ptx ` p1 ´ tqbq ˇ ˇ q dt ˙1_q .

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Since |f2

|q is s-convex in the second sense, we have ż1 0 ˇ ˇf2ptx ` p1´qaq ˇ ˇ q dt ≤ ż1 0 “tsˇ ˇf2pxq ˇ ˇ q ` p1 ´ tqsˇˇf2paq ˇ ˇ q_{‰ dt} “ |f 2 pxq|q` |f2paq|q s ` 1 and ż1 0 ˇ ˇf2ptx ` p1 ´ tqbq ˇ ˇ q dt ≤ ż1 0 “tsˇ ˇf2pxq ˇ ˇ q ` p1 ´ tqsˇˇf2pbq ˇ ˇ q_{‰ dt} “ |f 2_pxq|q_{` |f}2_pbq|q s ` 1 . Therefore, we have ˇ ˇ ˇ ˇ 1 b ´ a żb a f puqdu ´ f pxq ` ˆ x ´a ` b 2 ˙ f1pxq ˇ ˇ ˇ ˇ ≤ px ´ aq 3 2pb ´ aq ˆ 1 2p ` 1 ˙1 pˆ_|f2_pxq|q_{` |f}2_paq|q s ` 1 ˙1 q `pb ´ xq 3 2pb ´ aq ˆ 1 2p ` 1 ˙1 pˆ_|f2_pxq|q_{` |f}2_pbq|q s ` 1 ˙1_q , where 1_p `1_q “ 1, which is required.

Corollary 4. Under the above assumptions, we have the following in-equality: (2.8) ˇ ˇ ˇ ˇ 1 b ´ a żb a f puqdu ´ f pxq ` ˆ x ´a ` b 2 ˙ f1 pxq ˇ ˇ ˇ ˇ ≤ 3M p2p ` 1q 1 p ˆ 2 s ` 1 ˙1 q « pb ´ aq2 24 ` 1 2 ˆ x ´a ` b 2 ˙2ff . This follows by Theorem 5 with M “ sup

xPra,bs |f2| .

Corollary 5. With the assumptions in Corollary 4, one has the mid-point inequality: ˇ ˇ ˇ ˇ 1 b ´ a żb a f puqdu ´ fˆ a ` b 2 ˙ˇ ˇ ˇ ˇ ≤ pb ´ aq 2 8 p2p ` 1q1p ˆ 2 s ` 1 ˙1 q M. This follows by Corollary 4, choosing x “ a`b₂ .

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Corollary 6. With the assumptions in Corollary 4, one has the following perturbed trapezoid like inequality:

ˇ ˇ ˇ ˇ żb a f puqdu ´ pb ´ aq 2 rf paq ` f pbqs ` pb ´ aq2 4 `f 1 pbq ´ f1paq˘ ˇ ˇ ˇ ˇ ≤ pb ´ aq 3 2 p2p ` 1qp1 ˆ 2 s ` 1 ˙1 q M.

This follows using Corollary 4 with x “ a, x “ b, adding the results and using the triangle inequality for the modulus.

Theorem 6. Let I Ă r0, 8q, f : I Ñ R be a twice differentiable function on I˝ _{such that f}2 _{P L}

1ra, bs where a, b P I with a ă b. If |f2|q is s-convex in the second sense on ra, bs, for some fixed s P p0, 1s and q ≥ 1, then the following inequality holds:

Proof. Suppose that q ≥ 1. From Lemma 1 and by the well known power mean inequality, we have

ˇ ˇ ˇ ˇ 1 b ´ a żb a f puqdu ´ f pxq ` ˆ x ´a ` b 2 ˙ f1pxq ˇ ˇ ˇ ˇ ≤ px ´ aq 3 2pb ´ aq ż1 0 t2ˇˇf2ptx ` p1 ´ tqaq ˇ ˇdt ` pb ´ xq3 2pb ´ aq ż1 0 t2ˇˇf2ptx ` p1 ´ tqbq ˇ ˇdt ≤ px ´ aq 3 2pb ´ aq ˆż1 0 t2dt ˙1´1_qˆż1 0 t2ˇˇf2ptx ` p1 ´ tqaq ˇ ˇ q dt ˙1_q `pb ´ xq 3 2pb ´ aq ˆż1 0 t2dt ˙1´1_q ˆż1 0 t2ˇˇf2ptx ` p1 ´ tqbq ˇ ˇ q dt ˙1_q .

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Since |f2

|q is s-convex in the second sense, we have ż1 0 t2ˇˇf2ptx ` p1 ´ tqaq ˇ ˇqdt ≤ ż1 0 “ts`2ˇ ˇf2pxq ˇ ˇq` t2p1 ´ tqs ˇ ˇf2paq ˇ ˇq‰ dt “ |f 2_pxq|q s ` 3 ` 2 |f2_paq|q ps ` 1q ps ` 2q ps ` 3q and ż1 0 t2ˇˇf2ptx ` p1 ´ tqbq ˇ ˇ q dt ≤ ż1 0 “ts`2ˇ ˇf2pxq ˇ ˇ q ` t2p1 ´ tqsˇˇf2pbq ˇ ˇ q_{‰ dt} “ |f 2 pxq|q s ` 3 ` 2 |f2 pbq|q ps ` 1q ps ` 2q ps ` 3q. Therefore, we have ˇ ˇ ˇ ˇ 1 b ´ a żb a f puqdu ´ f pxq ` ˆ x á ` b 2 ˙ f1pxq ˇ ˇ ˇ ˇ ≤ px ´ aq 3 2pb ´ aq ˆ 1 3 ˙1´1 qˆ_|f2_pxq|q s ` 3 ` 2 |f2 paq|q ps ` 1q ps ` 2q ps ` 3q ˙1 q `pb ´ xq 3 2pb ´ aq ˆ 1 3 ˙1´1_q ˆ |f2pxq|q s ` 3 ` 2 |f2_pbq|q ps ` 1q ps ` 2q ps ` 3q ˙1 q . Corollary 7. Under the above assumptions we have the following inequal-ity ˇ ˇ ˇ ˇ 1 b ´ a żb a f puqdu ´ f pxq ` ˆ x á ` b 2 ˙ f1pxq ˇ ˇ ˇ ˇ ≤ M ˜ 3`s2` 3s ` 4˘ ps ` 1qps ` 2qps ` 3q ¸1 q « pb ´ aq2 24 ` 1 2 ˆ x á ` b 2 ˙2ff . This follows by Theorem 6 with M “ sup

xPra,bs |f2| .

Corollary 8. With the assuptions in Corollary 7, one has the mid-point inequality: ˇ ˇ ˇ ˇ 1 b ´ a żb a f puqdu ´ fˆ a ` b 2 ˙ˇ ˇ ˇ ˇ ≤ M ˜ 3`s2_{` 3s ` 4}˘ ps ` 1qps ` 2qps ` 3q ¸1 q pb ´ aq2 24 . This follows by Corollary 7, choosing x “ a`b₂ .

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Remark 2. If in Corollary 8 we choose s “ 1 and q “ 1, then we have the following inequality: ˇ ˇ ˇ ˇ 1 b ´ a żb a f puqdu ´ fˆ a ` b 2 ˙ˇ ˇ ˇ ˇ ≤ Mpb ´ aq 2 24 , which is the inequality (1.3) for functions f with convex |f2_{| .}

Corollary 9. With the assumptions in Corollary 7, one has the following perturbed trapezoid like inequality:

ˇ ˇ ˇ ˇ żb a f puqdu ´ pb ´ aq 2 rf paq ` f pbqs ` pb ´ aq2 4 `f 1 pbq ´ f1paq˘ ˇ ˇ ˇ ˇ ≤ pb ´ aq 3 6 ˜ 3`s2` 3s ` 4˘ ps ` 1qps ` 2qps ` 3q ¸1 q M.

This follows by using Corollary 7 with x “ a, x “ b, adding the results and using the triangle inequality for the modulus.

Remark 3. All of the above inequalities hold for functions f with convex |f2|. Simply choose s “ 1 in each of those results to get desired formulas.

The following result holds in the s-concave case.

Theorem 7. Let I Ă r0, 8q, f : I Ñ R be a twice differentiable function on I˝ _{such that f}2 _{P L}

1ra, bs where a, b P I with a ă b. If |f2|q is s-concave in the second sense on ra, bs, for some fixed s P p0, 1s, p, q ą 1 and 1_p`1_q “ 1, then the following inequality holds:

(2.10) ˇ ˇ ˇ ˇ 1 b ´ a żb a f puqdu ´ f pxq ` ˆ x ´a ` b 2 ˙ f1pxq ˇ ˇ ˇ ˇ ≤ 2 ps´1q{q p2p ` 1q1{ppb ´ aq ¨ ˚ ˚ ˝ px ´ aq3 ˇ ˇ ˇ ˇ f2 px ` a 2 q ˇ ˇ ˇ ˇ` pb ´ xq 3 ˇ ˇ ˇ ˇ f2 pb ` x 2 q ˇ ˇ ˇ ˇ 2 ˛ ‹ ‹ ‚ ,

for each x P ra, bs.

Proof. Suppose that q ą 1. From Lemma 1 and by the Hölder inequality, we have

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ˇ ˇ ˇ ˇ 1 b ´ a żb a f puqdu ´ f pxq ` ˆ x ´a ` b 2 ˙ f1pxq ˇ ˇ ˇ ˇ ≤ px ´ aq 3 2pb ´ aq ż1 0 t2ˇˇf2ptx ` p1 ´ tqaq ˇ ˇdt ` pb ´ xq3 2pb ´ aq ż1 0 t2ˇˇf2ptx ` p1 ´ tqbq ˇ ˇdt ≤ px ´ aq 3 2pb ´ aq ˆż1 0 t2pdt ˙1_pˆż1 0 ˇ ˇf2ptx ` p1 ´ tqaq ˇ ˇ q dt ˙1_q `pb ´ xq 3 2pb ´ aq ˆż1 0 t2pdt ˙1_pˆż1 0 ˇ ˇf2ptx ` p1 ´ tqbq ˇ ˇ q dt ˙1_q .

Since |f2_|q _{is s-concave in the second sense, using (1.1) we obtain} (2.11) ż1 0 ˇ ˇf2ptx ` p1 ´ tqaq ˇ ˇ q dt ≤ 2s´1 ˇ ˇ ˇ ˇ f2px ` a 2 q ˇ ˇ ˇ ˇ q and (2.12) ż1 0 ˇ ˇf2ptx ` p1 ´ tqbq ˇ ˇ q dt ≤ 2s´1 ˇ ˇ ˇ ˇ f2 pb ` x 2 q ˇ ˇ ˇ ˇ q . A combination of (2.11) and (2.12) gives

ˇ ˇ ˇ ˇ 1 b ´ a żb a f puqdu ´ f pxq ` ˆ x ´a ` b 2 ˙ f1 pxq ˇ ˇ ˇ ˇ ≤ 2 ps´1q{q p2p ` 1q1{ppb ´ aq ¨ ˚ ˚ ˝ px ´ aq3 ˇ ˇ ˇ ˇf 2 px ` a 2 q ˇ ˇ ˇ ˇ` pb ´ xq 3 ˇ ˇ ˇ ˇf 2 pb ` x 2 q ˇ ˇ ˇ ˇ 2 ˛ ‹ ‹ ‚ .

This completes the proof.

Corollary 10. If in (2.10), we choose x “ a`b2 , then we have (2.13) ˇ ˇ ˇ ˇ 1 b ´ a żb a f puqdu ´ fˆ a ` b 2 ˙ˇ ˇ ˇ ˇ ≤ 2 ps´1q{q pb ´ aq2 16 p2p ` 1q1{p „ˇ ˇ ˇ ˇ f2 p3a ` b 4 q ˇ ˇ ˇ ˇ` ˇ ˇ ˇ ˇ f2 pa ` 3b 4 q ˇ ˇ ˇ ˇ  . For instance, if s “ 1, then we have

ˇ ˇ ˇ ˇ 1 b ´ a żb a f puqdu ´ fˆ a ` b 2 ˙ˇ ˇ ˇ ˇ ≤ pb ´ aq 2 16 p2p ` 1q1{p „ˇ ˇ ˇ ˇ f2 p3a ` b 4 q ˇ ˇ ˇ ˇ` ˇ ˇ ˇ ˇ f2 pa ` 3b 4 q ˇ ˇ ˇ ˇ  .

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E. Set

DEPARTMENT OF MATHEMATICS FACULTY OF SCIENCE AND ARTS ORDU UNIVERSITY

ORDU, TURKEY

E-mail: erhanset@yahoo.com

M. Z. Sarikaya DEPARTMENT OF MATHEMATICS FACULTY OF SCIENCE AND ARTS DÜZCE UNIVERSITY DÜZCE, TURKEY E-mail: sarikayamz@gmail.com M. E. Ozdemir ATATÜRK UNIVERSITY K.K. EDUCATION FACULTY DEPARTMENT OF MATHEMATICS 25240, CAMPUS, ERZURUM, TURKEY E-mail: emos@atauni.edu.tr