Research Article Hermite–Hadamard Type Inequalities via Generalized Harmonic Exponential Convexity and Applications Saad Ihsan Butt , 1 Muhammad Tariq, 1 Adnan Aslam , 2 Hijaz Ahmad , 3 and Taher A. Nofal 4 1 COMSATS University Islamabad, Lahore Campus, Pakistan 2 Department of Natural Sciences and Humanities, University of Engineering and Technology, Lahore (RCET), Pakistan 3 Department of Basic Sciences, University of Engineering and Technology, Peshawar, Pakistan 4 Department of Mathematic, College of Science, Taif University, P.O. Box 11099, Taif 21944, Saudi Arabia Correspondence should be addressed to Hijaz Ahmad; hijaz555@gmail.com Received 5 January 2021; Revised 23 January 2021; Accepted 31 January 2021; Published 12 February 2021 Academic Editor: Gangadharan Murugusundaramoorthy Copyright © 2021 Saad Ihsan Butt 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 work, we introduce the idea and concept of m–polynomial p–harmonic exponential type convex functions. In addition, we elaborate the newly introduced idea by examples and some interesting algebraic properties. As a result, several new integral inequalities are established. Finally, we investigate some applications for means. The amazing techniques and wonderful ideas of this work may excite and motivate for further activities and research in the different areas of science. 1. Introduction Theory of convexity present an active, fascinating, and attractive field of research and also played prominence and amazing act in different fields of science, namely, mathemat- ical analysis, optimization, economics, finance, engineering, management science, and game theory. Many researchers endeavor, attempt, and maintain his work on the concept of convex functions and extend and generalize its variant forms in different ways using innovative ideas and fruitful techniques. Convexity theory provides us with a unified framework to develop highly efficient, interesting, and pow- erful numerical techniques to tackle and to solve a wide class of problems which arise in pure and applied sciences. In recent years, the concept of convexity has been improved, generalized, and extended in many directions. The concept of convex functions also played prominence and meaningful act in the advancement of the theory of inequalities. A num- ber of studies have shown that the theory of convex functions has a close relationship with the theory of inequalities. The integral inequalities are useful in optimization the- ory, functional analysis, physics, and statistical theory. In diverse and opponent research, inequalities have a lot of applications in statistical problems, probability, and numeri- cal quadrature formulas [1–3]. So eventually due to many generalizations, variants, extensions, widespread views, and applications, convex analysis and inequalities have become an attractive, interesting, and absorbing field for the researchers and for attention; the reader can refer to [4–6]. Recently Kadakal and Iscan [7] introduced a generalized form of convexity, namely, n–polynomial convex functions. It is well known that the harmonic mean is the special case of power mean. It is often used for the situations when the average rates is desired and have a lot of applications in different field of sciences which are statistics, computer sci- ence, trigonometry, geometry, probability, finance, and elec- tric circuit theory. Harmonic mean is the most appropriate measure for rates and ratios because it equalizes the weights of each data point. Harmonic mean is used to define the har- monic convex set. In 2003, first time harmonic convex set was introduced by Shi [8]. Harmonic and p–harmonic con- vex function was for the first time introduced and discussed by Anderson et al. [9] and Noor et al. [10], respectively. Awan et al. [11] keeping his work on generalizations, Hindawi Journal of Function Spaces Volume 2021, Article ID 5533491, 12 pages https://doi.org/10.1155/2021/5533491