Ostrowski type inequalities via some exponentially convex functions with applications

1.   Introduction

Let $ \curlyvee:[h_1, h_2]\rightarrow\mathbb{R} $ be a differentiable mapping on $ (h_1, h_2) $ whose derivative $ \curlyvee':((h_1, h_2)\rightarrow\mathbb{R} $ is bounded on $ (h_1, h_2) $, i.e., $ |\curlyvee'(l)|\leq M $ for all $ l\in (h_1, h_2) $. Then the following inequality holds:

for all $ l\in [h_1, h_2] $. Many authors find the inequality (1.1) for other generalized convex functions. For more results and details see ^{[1,3,4,5,6,8,9,10]}.

Awan et al. ^[2] introduced following new class of convex functions.

Definition 1.1. (^[2]) A function $ \curlyvee: \mathcal{H}\subseteq \mathbb{R}\rightarrow \mathbb{R} $ is called exponentially convex, if

for all $ h_1, h_2\in \mathcal{H} $, $ \tau\in[0, 1] $ and $ \alpha\in \mathbb{R} $. If the inequality (1.2) is in reversed order then $ \curlyvee $ is called exponentially concave.

Mehreen and Anwar ^[7] introduced an other class of functions as:

Definition 1.2. (^[7]) Let $ s\in(0, 1] $ and $ \mathcal{H}\subset \mathbb{R}_{0} $ be an interval. A function $ \curlyvee:\mathcal{H}\rightarrow \mathbb{R} $ is called exponentially $ s $-convex in the second sense, if

for all $ h_1, h_2\in \mathcal{H} $, $ \tau\in[0, 1] $ and $ \alpha\in \mathbb{R} $. If (1.3) is in reversed order then $ \curlyvee $ is called exponentially $ s $-concave.

Example 1.1. A function $ \curlyvee:[0, \infty)\rightarrow \mathbb{R} $, defined by $ \curlyvee(l) = \ln(l) $ for $ s\in(0, 1) $ is an exponentially $ s $-convex in the second sense, for all $ \alpha\leq -1 $.

Mehreen and Anwar ^[7] proved following Hadamard's inequality for exponentially $ s $-convex in second sense.

Theorem 1.2. (^[7]) Let $ \curlyvee:\mathcal{H}\subset \mathbb{R}_{0}\rightarrow \mathbb{R} $ be an integrable exponentially $ s $-convex function in the second sense on $ \mathcal{H}^\circ $. Then for $ h_1, h_2\in \mathcal{H} $ with $ h_1 < h_2 $ and $ \alpha\in\mathbb{R} $, we have

where

2.   Ostrowski type inequalities

First we state the following lemma given in ^[3].

Lemma 2.1. (^[3]) Let $ \curlyvee:\mathcal{H}\rightarrow \mathbb{R} $ be a differentiable mapping on $ \mathcal{H}^\circ $ for $ h_1, h_2\in \mathcal{H} $ with $ h_1 < h_2 $. If $ \curlyvee'\in L_1[h_1, h_2] $, then the following equality holds:

for each $ l\in [h_1, h_2] $.

Now we prove the following theorem.

Theorem 2.1. Let $ \curlyvee:\mathcal{H}\rightarrow \mathbb{R} $ be a differentiable mapping on $ \mathcal{H}^\circ $ such that $ \curlyvee'\in[h_1, h_2] $ for $ h_1, h_2\in \mathcal{H} $ with $ h_1 < h_2 $. If $ \curlyvee' $ is exponentially $ s $-convex in second sense on $ [h_1, h_2] $ for some $ s\in(0, 1] $ and $ |\curlyvee'(l)|\leq M $, $ l\in [h_1, h_2] $, then the following inequality holds:

for each $ l\in [h_1, h_2] $.

Proof. Using Lemma 2.1 and since $ \curlyvee' $ is exponentially $ s $-convex, we have

Since

This completes the proof.

Remark 2.1. In Theorem $ 2.1 $, by letting $ \alpha = 0 $, we get inequality $ 2.1 $ of Theorem $ 2 $ in ^[1].

Corollary 2.1. Under the similar considerations of Theorem $ 2.1 $, by taking $ s = 1 $, we get

Remark 2.2. In Corollary $ 2.1 $, by letting $ \alpha = 0 $, we get inequality $ (1.1) $.

Theorem 2.2. Let $ \curlyvee:\mathcal{H}\rightarrow \mathbb{R} $ be a differentiable mapping on $ \mathcal{H}^\circ $ such that $ \curlyvee'\in[h_1, h_2] $ for $ h_1, h_2\in \mathcal{H} $ with $ h_1 < h_2 $. If $ |\curlyvee'|^q $ is exponentially $ s $-convex in the second sense on $ [h_1, h_2] $ for some $ s\in(0, 1] $, $ p $, $ q > 1 $, $ \frac{1}{p}+\frac{1}{q} = 1 $ and $ |\curlyvee'(l)|\leq M $, $ l\in [h_1, h_2] $, then the following inequality holds:

for each $ l\in [h_1, h_2] $.

Proof. Let $ p > 1 $. From Lemma 2.1 and using H$ \ddot{o} $lder's inequality, we obtain

Since $ |\curlyvee'|^q $ is exponentially $ s $-convex in the second sense and $ |\curlyvee'(l)|\leq M $, then we have

and

Hence using $ (2.7) $ and $ (2.8) $ in $ (2.6) $, we get $ (2.5) $

Remark 2.3. In Theorem $ 2.2 $, by letting $ \alpha = 0 $, we get inequality $ 2.2 $ of Theorem $ 3 $ in ^[1].

Corollary 2.2. Under the assumptions of Theorem $ 2.2 $, by taking $ s = 1 $, we get

Remark 2.4. In Corollary 2.2, by letting $ \alpha = 0 $, we get result for convex function.

Theorem 2.3. Let $ \curlyvee:\mathcal{H}\rightarrow \mathbb{R} $ be a differentiable mapping on $ \mathcal{H}^\circ $ such that $ \curlyvee'\in[h_1, h_2] $ for $ h_1, h_2\in \mathcal{H} $ with $ h_1 < h_2 $. If $ |\curlyvee'|^q $ is exponentially $ s $-convex in second sense on $ [h_1, h_2] $ for some $ s\in(0, 1] $, $ q\geq 1 $ and $ |\curlyvee'(l)|\leq M $, $ l\in [h_1, h_2] $, then the following inequality holds:

for each $ l\in [h_1, h_2] $.

Proof. Let $ p > 1 $. From Lemma 2.1 and using power mean inequality, we obtain

Since $ |\curlyvee'|^q $ is exponentially $ s $-convex in the second sense and $ |\curlyvee'(l)|\leq M $, then we have

and

Hence using $ (2.12) $ and $ (2.13) $ in $ (2.11) $, we get $ (2.10) $

Remark 2.5. In Theorem $ 2.3 $, by letting $ \alpha = 0 $, we get inequality $ (2.3) $ of Theorem $ 4 $ in ^[1].

Corollary 2.3. Under the assumptions of Theorem $ 2.3 $, by taking $ s = 1 $, we get

Remark 2.6. In Corollary 2.3, by letting $ \alpha = 0 $, we get result for convex function.

Remark 2.7. From previous inequalities, we can obtain several midpoint type inequalities by setting $ l = \frac{h_1+h_2}{2} $. However, the details are omitted for interested reader.

Theorem 2.4. Let $ \curlyvee:\mathcal{H}\rightarrow \mathbb{R} $ be a differentiable mapping on $ \mathcal{H}^\circ $ such that $ \curlyvee'\in[h_1, h_2] $ for $ h_1, h_2\in \mathcal{H} $ with $ h_1 < h_2 $. If $ |\curlyvee'|^q $ is exponentially $ s $-concave on $ [h_1, h_2] $, $ p $, $ q > 1 $, $ \frac{1}{p}+\frac{1}{q} = 1 $, then the following inequality holds:

for each $ l\in [h_1, h_2] $.

Proof. Let $ p > 1 $. From Lemma 2.1 and using H$ \ddot{o} $lder's inequality, we obtain

Since $ |\curlyvee'| $ is exponentially $ s $-concave and from inequality (1.4), we have

and

Using inequalities (2.18) and (2.17) in (2.16), we get (2.15).

Remark 2.8. If one takes $ l = \frac{h_1+h_2}{2} $ in (2.15) then one gets inequality $ (2.8) $ in ^[1].

Similarly, by letting $ s = 1 $ in (2.15) then one gets inequality $ (2.9) $ in ^[1].

3.   Applications to special means

Consider some special means of two positive numbers $ h_1, h_2 $, $ h_1 < h_2 $:

$ {\rm (1).} $ The arithmetic mean

$ {\rm (2).} $ The Identric mean

where $ h_1, h_2 > 0 $.

$ {\rm (3).} $ The $ p $-logarithmic mean

Proposition 3.1. Let $ 0 < h_1 < h_2 $, $ q\geq 1 $ and $ 0 < s < 1 $. Then we have

Proof. the result holds by letting $ l = \frac{h_1+h_2}{2} $ in (2.10) with exponentially $ s $-convex function in second sense $ \curlyvee:(0, \infty)\rightarrow \mathbb{R} $, $ \curlyvee(l) = l^s $ for all $ \alpha\leq -1 $.

Proposition 3.2. Let $ 0 < h_1 < h_2 $, $ q\geq 1 $ and $ 0 < s < 1 $. Then we have

Proof. the result holds by letting $ l = \frac{h_1+h_2}{2} $ in (2.10) with exponentially $ s $-convex function in second sense $ \curlyvee:(0, \infty)\rightarrow \mathbb{R} $, $ \curlyvee(l) = \ln(l) $ for all $ \alpha\leq -1 $.

Remark 3.1. From Corollary 2.3 one can gets more applications to some special means for exponentially convex functions.

4.   Conclusion

From Corollaries 2.1, 2.2 and 2.3, we obtained new ostrowski's type inequalities for exponentially convex function. From Theorems 2.1, 2.2, 2.3 and 2.4, we obtained new ostrowski's type inequalities for exponentially $ s $-convex function. Some applications are obtained from Propositions 3.1 and 3.2.

Acknowledgement

This research article is supported by National University of Sciences and Technology(NUST), Islamabad, Pakistan.

Conflict of interest

The authors declare that there is no interest regarding the publication of this paper.