Integral inequalities and the Mittag-Leffler function play a crucial role in many branches of mathematics and applications, including fractional calculus, mathematical physics, and engineering. In this paper, we introduced an extended generalized Mittag-Leffler function that involved several well-known Mittag-Leffler functions as a special case. We also introduced an associated generalized fractional integral to obtain some estimates for fractional integral inequalities of the Hermite-Hadamard and Hermite-Hadamard-Fejér types. This article offered several analytical tools that will be useful to anyone working in this field. To demonstrate the veracity of our findings, we offered a few numerical and graphical examples. A few applications of modified Bessel functions and unitarily invariant norm of matrices were also given.

In this paper, we define a new class ${\mathcal{R}}_{t,\delta ,\upsilon }^{m,n,\sigma }\left(\mathcal{A},\mathcal{B}\right)$ of holomorphic functions in the open unit disk defined connected with the combination binomial series and Babalola operator using the differential subordination with Janowski-type functions. Using the well-known Carathéodory's inequality for function with real positive parts and the Keogh-Merkes and Ma-Minda's in equalities, we determined the upper bound for the first two initial coefficients of the Taylor-Maclaurin power series expansion. Then, we found an upper bound for the Fekete-Szegö functional for the functions in this family. Further, a similar result for the first two coefficients and for the Fekete-Szegő inequality have been done the function ${\mathcal{G}}^{-1}$ when $\mathcal{G}\in {\mathcal{R}}_{t,\delta ,\upsilon }^{m,n,\sigma }\left(\mathcal{A},\mathcal{B}\right)$. Next, for the functions of these newly defined family we determine coefficient estimates, distortion bounds, radius problems, and the radius of starlikeness and close-to-convexity. The novelty of the results is that we were able to investigate basic properties of these new classes of functions using simple methods and these classes are connected with the new convolution operator and the Janowski functions.

Integral inequalities in mathematical interpretations are a substantial and ongoing body of research. Because fractional calculus techniques are widely used in science, a lot of research has recently been done on them. A key concept in fractional calculus is the Caputo-Fabrizio fractional integral. In this work, we focus on using the Caputo-Fabrizio fractional integral operator to build a multi-parameter fractional integral identity. Using the obtained integral identity, certain generalized estimates of Bullen-type fractional inequalities have been generated. By establishing certain inequalities, this study advances the fields of fractional calculus and convex function research. Both graphical and numerical statistics are provided to show the correctness of our results. We finally provide applications to modified Bessel functions, $\mathfrak{h}$-divergence measures, and probability density functions.

In this paper, we introduced a novel norm structure and corresponding modular for Orlicz-Zygmund spaces, designed to capture summability and integrability properties under non-standard growth conditions. Moreover, we established the Hermite-Hadamard inequalities and gave a positive answer to the open problem 3.11 of Almeida and Hästö in their article (Besov spaces with variable smoothness and integrability, J. Funct. Anal., 258 (2010), 1628–1655), since the Hermite-Hadamard inequalities improve triangular-type inequalities. In addition, we explored some new structural properties of Besov-type Morrey spaces of summability and integrability. Furthermore, we addressed an open problem concerning the compactness of Morrey-type spaces characterized by summability and integrability properties. This problem was originally posed by Peter Hästö during the conference "Nonstandard Growth Phenomena", held in Turku, Finland, from August 29 to 31, 2017.