Certain Hermite – Hadamard Inequalities for Logarithmically Convex Functions with Applications

Shilpi Jain 1,* , Khaled Mehrez 2 ,3, Dumitru Baleanu 4 and Praveen Agarwal 5,6 1 Department of Mathematics, Poornima College of Engineering, Jaipur 302022, India 2 Département de Mathématiques, Facultée des sciences de Tunis, Université Tunis El Manar, Tunis 1068, Tunisia; k.mehrez@yahoo.fr 3 Dèpartement de Mathématiques, Issat Kasserine, Université de Kairouan, Kairouan 3100, Tunisia 4 Department of Mathematics and Computer Sciences, Faculty of Art and Sciences, Çankaya University, Balgat 0630, Turkey; dumitru.baleanu@gmail.com 5 Department of Mathematics, Anand International College of Engineering, Jaipur 303012, India; goyal.praveen2011@gmail.com 6 Department of Mathematics, Harish-Chandra Research Institute (HRI), Allahbad 211019, India * Correspondence: shilpijain1310@gmail.com


Introduction
In recent years, because of the importance of the relationship between inequalities and convexity, the study of the Hermite-Hadamard's inequality has attracted much attention due to applications of convex functions (for details see [1,2]).
A function f : I ⊆ R −→ R is convex.Then, for a, b ∈ I, a < b, we have A function f : [a, b] ⊆ R → (0, ∞) is log-convex.Then, for x, y ∈ [a, b] and t ∈ [0, 1], we have If the above inequality is reversed then f is called a log-concave function.It is also known that if g is differentiable, then f is log-convex (log-concave) if and only if f / f is increasing (decreasing).It is easy to see this by the arithmetic-mean-geometric-mean (AM-GM) inequality for all x, y ∈ I and t ∈ [0, 1], i.e., the set of log-convex functions is a proper subset of the class of convex functions.
The main idea of this paper is motivated by the above results.The paper is organized as follows.In Section 2, we first present some lemmas which are used in the main results.Then, we present some estimates to the left-hand side and right-hand side of Hermite-Hadamard inequalities for functions whose absolute values of the first and second derivatives to positive real powers are log-convex.In Section 3, we derive some inequalities for q-digamma and q-polygamma functions by applying the main results of Section 2 and deducing some of them for the q-analogue harmonic numbers.Moreover, several inequalities for special means are derived.In Section 3, our aim is to apply the idea taken in the previous section for the q-digamma and q-polygamma functions and the classical Hermite-Hadamard inequalities (1).As applications, we show new lower and upper bounds for the q-analogue of the harmonic numbers in terms of the q-trigamma function.Moreover, some inequalities for the harmonic numbers and Euler-Mascheroni constant are presented.

Main Results
In this section, we establish our main results, but first we briefly recall some basic well-known results which will be used in the sequel.
Theorem 1.Let f : where Proof.Let q = 1 and by means of Lemma 1 and known result [5] (p.4), we have where α, β > 1 such that 1 α + 1 β = 1, we find that Now, by using the following formula | in I 11 and I 12 , respectively, we get , and Next by considering the Hölder-Rogers inequality, for q , p = q q −1 , we get β dt , hence, we get (5).

Remark 1.
In view of the inequalities ( 5) and (8) when | f (a)| = 1, we deduced that the inequality (5) reduces to In the case when A similar result is embodied in this theorem.The beta function and gamma function are defined by The beta function satisfies the following properties: In particular, where p = q q −1 and α, β > 1 such that Proof.By Lemma 1, we have So, by using the Hölder-Rogers inequality (9) and the arithmetic-mean-geometric-mean (AM-GM) inequality ( 6), we have The desired inequality (13) is thus established.
then the inequality (19) reduces to the following inequality: Moreover, in the case when holds for all p, q , α, β > 1 such that 1 p + 1 q = 1 and Proof.From Lemma 2, we get On the other hand, by using the Hölder-Rogers inequality (9) and the AM-GM inequality (6), we obtain Similarly, we get which implies (19).26) and (17), we get Proof.Since | f | is, in fact, log-convex and due to the AM-GM inequality (6), we successively get So, along with the identity we get Also from the AM-GM inequality (6), and since the function | f | is log-convex, we get Taking the values we deduce In view of (36), ( 38) and (39), we can easily get the following inequality Taking into account Lemma 2 along with (35) and (40), we obtain (30).17) and (30), we obtain 3. Applications

Applications to Special Means
Here, we demonstrate new inequalities connecting the above means for arbitrary real numbers.
As another application of inequality (46) we can provide the following inequalities for the q-triagamma and q-polygamma functions and the q-analogue of harmonic numbers H nq defined by So, in view of inequality (46) and using the equation we obtain the following result.
Corollary 1.Let n ∈ N and 0 < q < 1.Then, the following inequality holds true.

Proposition 5.
Let n be a integer number.Then the inequality holds, where H n is a harmonic number.
Remark 6.By using the equation where γ is the Euler-Mascheroni constant, the inequality (50) can be read as   |ψ (2) (1)| − 1 log |ψ (2) (1)| 2 + |ψ (2) (n + 1)| − 1 log |ψ (2) (n We conclude our investigation by applying the methods which were developed in Section 2, for the classical Hermite-Hadamard inequalities.Since the q-trigamma function ψ q (x) is completely monotonic on (0, ∞) for each q ∈ (0, 1), and consequently is convex on (0, ∞).Then we get In particular, we obtain the following inequalities for the q-analogue of harmonic numbers Letting q −→ 1, in the above inequalities, we obtain the new inequality for the harmonic number H n , Now, combining the inequalities (56) and (52), we get the two-sided bounding inequalities for the Euler-Mascheroni constant: Author Contributions: All authors contributed equally.
Funding: No specific funding was received for this project.

Remark 4 .
Let the assumptions of Theorem 4 be satisfied, with | f (x)| > 1 for all x ∈ [a, b].In view of (