Certain Hermite–Hadamard Inequalities for Logarithmically Convex Functions with Applications

In this paper, we discuss various estimates to the right-hand (resp. left-hand) side of the Hermite–Hadamard inequality for functions whose absolute values of the second (resp. first) derivatives to positive real powers are log-convex. As an application, we derive certain inequalities involving the q-digamma and q-polygamma functions, respectively. As a consequence, new inequalities for the q-analogue of the harmonic numbers in terms of the q-polygamma functions are derived. Moreover, several inequalities for special means are also considered.

Coefficients of strongly alpha-convex and alpha-logarithmicaly convex functions

Let the function $f$ be analytic in $D=\{z:|z|<1\}$ and be given by $f(z)=z+\sum_{n=2}^{\infty}a_{n}z^{n}$. For $0< \beta \le 1$, denote by $C (\beta)$ and $S^*(\beta)$ the classes of strongly convex functions and strongly starlike functions respectively. For $0\le \alpha \le1$ and $0< \beta \le 1$, let $M(\alpha, \beta)$ be the class of strongly alpha-convex functions defined by $\left|\arg \Big((1-\alpha) \dfrac{zf'(z)}{f(z)}\Big)+\alpha (1+\dfrac{zf''(z)}{f'(z)})^{}\Big)\right|< \dfrac{\pi \beta }{2}$, and $M^{*}(\alpha, \beta)$ the class of strongly alpha-logarithmically convex functions defined by $\left|\arg\Big( \Big( \dfrac{zf'(z)}{f(z)}\Big)^{1-\alpha}\Big(1+\dfrac{zf''(z)}{f'(z)}\Big)^{\alpha}\Big)\right|< \dfrac{\pi \beta }{2}$. We give sharp bounds for the initial coefficients of $f\in M(\alpha,\beta)$ and $f\in M^{*}(\alpha,\beta)$, and for the initial coefficients of the inverse function $f^{-1}$ of $f\in M(\alpha,\beta)$ and $f\in M^{*}(\alpha,\beta)$. These results generalise and unify known coefficient inequalities for $C (\beta)$ and $S^*(\beta)$

On the second hankel determinant for the kth-root transform of analytic functions

Let f be a normalized analytic function in the open unit disk of the complex plane satisfying zf'(z)/f(z) is subordinate to a given analytic function ?. A sharp bound is obtained for the second Hankel determinant of the kth-root transform z[f(zk)/zk]1/k. Best bounds for the Hankel determinant are also derived for the kth-root transform of several other classes, which include the class of ?-convex functions and ?-logarithmically convex functions. These bounds are expressed in terms of the coefficients of the given function ?, and thus connect with earlier known results for particular choices of ?.

On Hermite-Hadamard type integral inequalities for n-times differentiable m- and (α,m)-logarithmically convex functions

In this paper, we establish Hermite-Hadamard type inequalities for functions whose nth derivatives are m- and (?;m)-logarithmically convex functions. From our results, several results for classical trapezoidal and classical midpoint inequalities are obtained in terms second derivatives that are m- and (?,m)-logarithmically convex functions as special cases.