Research Article A Refinement of the Integral Jensen Inequality Pertaining Certain Functions with Applications Zaid Mohammed Mohammed Mahdi Sayed , 1,2 Muhammad Adil Khan , 2 Shahid Khan, 2 and Josip Pečarić 3 1 Department of Mathematics, University of Sa’adah, Sa’adah 1872, Yemen 2 Department of Mathematics, University of Peshawar, Peshawar 25000, Pakistan 3 Department of Mathematical, Physical and Chemical Sciences, Croatian Academy of Sciences and Arts, Zrinski trg 11, 10000 Zagreb, Croatia Correspondence should be addressed to Zaid Mohammed Mohammed Mahdi Sayed; zaidmohamm56@gmail.com Received 7 March 2022; Accepted 28 June 2022; Published 30 July 2022 Academic Editor: Hugo Leiva Copyright © 2022 Zaid Mohammed Mohammed Mahdi Sayed et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. In this paper, we present a new refinement of the integral Jensen inequality by utilizing certain functions and give its applications to various means. We utilize the refinement to obtain some new refinements of the Hermite-Hadamard and Hölder’s inequalities as well. Also, we present its applications in information theory. At the end of this paper, we give a more general form of the proposed refinement of the Jensen inequality, associated to several functions. 1. Introduction Being an important part of modern applied analysis, the field of mathematical inequalities has recorded an exponential growth with significant impact on various parts of science and technology [1–5]. These inequalities are also extended and generalized in various aspects; one can see such results in [6–15]. The Jensen weighted integral inequality is a cen- tral tool among them; its basic form is follows as [16]. Theorem 1. Assume a convex function f : I ⟶ ℝ and g, h : ½θ 1 , θ 2  ⟶ ℝ are measurable functions such that gðθÞ ∈ I and hðθÞ ≥ 0∀θ ∈ ½θ 1 , θ 2 . Also, suppose that h, gh, ð f ∘ gÞ:h are all integrable functions on ½θ 1 , θ 2  and Ð θ 2 θ 1 hðθÞdθ > 0, then f Ð θ 2 θ 1 g θ ðÞh θ ðÞdθ Ð θ 2 θ 1 h θ ðÞdθ 0 @ 1 A ≤ Ð θ 2 θ 1 f ∘ g ð Þ θ ðÞh θ ðÞdθ Ð θ 2 θ 1 h θ ðÞdθ : ð1Þ The Jensen inequality is one of the fundamental inequalities in modern applied analysis. This inequality is of pivotal impor- tance because various other classical inequalities, for example, the Beckenbach-Dresher, Minkowski’s, the Hermite-Hadamard, Ky-Fan’s, Hölder’s, the arithmetic-geometric, and Levinson’s and Young’s inequalities, can be deduced from this inequality. Also, this inequality can be treated as a problem solving oriented tool in different areas of science and technology, and an extensive literature is dedicated to this inequality regarding its counterparts, generalizations, improvements, and converse results (see, for instance, [17–21]) and the refer- ences therein. The Hermite-Hadamard inequality is presented as follows ([22], page 10 in [23]). Theorem 2. Assume a convex function f : ½θ 1 , θ 2  ⟶ ℝ, then the following double inequalities hold: f θ 1 + θ 2 2   ≤ 1 θ 2 − θ 1 ð θ 2 θ 1 f θ ðÞdθ ≤ f θ 1 ð Þ + f θ 2 ð Þ 2 : ð2Þ Hindawi Journal of Function Spaces Volume 2022, Article ID 8396644, 11 pages https://doi.org/10.1155/2022/8396644