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.

Fractional calculus can be used to fully describe numerous real-world situations in a wide range of scientific disciplines, including natural science, social science, electrical, chemical, and mechanical engineering, economics, statistics, weather forecasting, and particularly biomedical engineering. Different types of derivatives can be useful when solving various fractional calculus problems. In this study, we suggested a single step modified one parameter family of the Caputo-type fractional iterative method. Convergence analysis shows that the proposed family of methods' order of convergence is $\vartheta +1$. To determine the error equation of the proposed technique, the computer algebra system CAS-Maple is employed. To illustrate the accuracy, validity, and usefulness of the proposed technique, we consider a few real-world applications from the fields of civil and chemical engineering. In terms of residual error, computational time, computational order of convergence, efficiency, and absolute error, the test examples' acquired numerical results demonstrate that the newly proposed algorithm performs better than the other classical fractional iterative scheme already existing in the literature. Using the computer program Mathematia 9.0, we compare the draw basins of attraction of the suggested fractional numerical algorithm to those of the currently used fractional iterative methods for the graphical analysis. The graphical results show how quickly the newly developed fractional method converges, confirming its supremacy to other techniques.

The main aim of this article is to study an extension of the Beta and Gamma matrix functions by using a two-parameter Mittag-Leffler matrix function. In particular, we investigate certain properties of these extended matrix functions such as symmetric relation, integral representations, summation relations, generating relation and functional relation.

A remarkably large number of unified integrals involving the Mittag–Leffler function have been presented. Here, with the same technique as Choi and Agarwal, we propose the establishment of two generalized integral formulas involving a multivariate generalized Mittag–Leffler function, which are expressed in terms of the generalized Lauricella series due to Srivastava and Daoust. We also present some interesting special cases.