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Here, we study the extension of p-trigonometric functions sinp and cosp family in complex domains and p-hyperbolic functions sinhp and the coshp family in hyperbolic complex domains. These functions satisfy analogous relations as their classical counterparts with some unknown properties. We show the relationship of these two classes of special functions viz. p-trigonometric and p-hyperbolic functions with imaginary arguments. We also show many properties and identities related to the analogy between these two groups of functions. Further, we extend the research bridging the concepts of hyperbolic and elliptical complex numbers to show the properties of logarithmic functions with complex arguments.

Convex functions play an integral part in artificial intelligence by providing mathematical guarantees that make optimization more efficient and reliable. In this manuscript, we originate and analyze a novel category of convexity, namely, harmonically trigonometric p ‐convex functions, and explore their properties. We provide examples of this new class of convex functions. By leveraging the new convexity, refinements of Hermite–Hadamard‐type and Fejér–Hermite–Hadamard‐type inequalities are formulated. The derivation of these inequalities involves the utilization of Hölder’s inequality, Hölder–İşcan inequality, the power‐mean integral inequality, and certain generalizations associated with these mathematical principles. The validity of the established results is confirmed through visual representation. A comparative analysis is provided to clarify that inequality derived through the power‐mean inequality is more refined than other inequalities. Additionally, we discuss the applications of these findings to some special means.

This article introduces (p,q)-analogs of the gamma integral operator and discusses their expansion to power functions, (p,q)-exponential functions, and (p,q)-trigonometric functions. Additionally, it validates other findings concerning (p,q)-analogs of the gamma integrals to unit step functions as well as first- and second-order (p,q)-differential operators. In addition, it presents a pair of (p,q)-convolution products for the specified (p,q)-analogs and establishes two (p,q)-convolution theorems.

In this paper, we mainly show some bounds and inequalities for the$ p $ -generalized trigonometric functions defined by Richter.

This paper constructs and introduces ( p , q ) -cosine and ( p , q ) -sine Bernoulli polynomials using ( p , q ) -analogues of ( x + a ) n . Based on these polynomials, we discover basic properties and identities. Moreover, we determine special properties using ( p , q ) -trigonometric functions and verify various symmetric properties. Finally, we check the symmetric structure of the approximate roots based on symmetric polynomials.

Utilizing p,q-numbers and p,q-concepts, in 2016, Duran et al. considered p,q-Genocchi numbers and polynomials, p,q-Bernoulli numbers and polynomials and p,q-Euler polynomials and numbers and provided multifarious formulas and properties for these polynomials. Inspired and motivated by this consideration, many authors have introduced (p,q)-special polynomials and numbers and have described some of their properties and applications. In this paper, using the (p,q)-cosine polynomials and (p,q)-sine polynomials, we consider a novel kinds of (p,q)-extensions of geometric polynomials and acquire several properties and identities by making use of some series manipulation methods. Furthermore, we compute the p,q-integral representations and p,q-derivative operator rules for the new polynomials. Additionally, we determine the movements of the approximate zerosof the two mentioned polynomials in a complex plane, utilizing the Newton method, and we illustrate them using figures.

In this paper, we introduce two novel classes of functions termed the harmonically trigonometric p -convex functions on Δ=[τ,υ]×[φ,χ] and harmonically trigonometric p -coordinated convex functions. We discuss relations between these two newly introduced classes of convex functions in two variables and validate the theoretical findings with visual 3D graphs that illustrate the relation landscapes and transition regimes between the function classes. We then study several special or limiting cases (e.g. p=1 , p=−1 , pure harmonic trigonometric and pure trigonometric), showing that our general formulations leads to new novel convexities. Hermite-Hadamard, Fejér-Hermite-Hadamard, and related type integral inequalities, with bounds including hypergeometric functions, are presented via a novel coordinated class of convex functions.