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In this paper, we introduce a new class of analytic functions, denoted by S(ν,φ[sub.ϑ,e]), and provide illustrative examples to elucidate its properties. This class generalizes the starlike and convex functions previously defined by Khatter et al. in relation to the exponential function. A significant contribution of this work is the derivation of sharp bounds for various coefficient-related problems within this class. The computational challenges involved in deriving these bounds were effectively addressed using Mathematica[sup.TM] codes. Additionally, figures illustrating the geometric properties and essential computations have been incorporated into the paper.

Let Se∗ denote the class of analytic functions f in the open unit disk normalized by f(0)=f′(0)-1=0 and satisfying the condition zf′(z)/f(z)≺ez for |z|<1. The structural formula, inclusion relations, coefficient estimates, growth and distortion results, subordination theorems and various radii constants for functions in the class Se∗ are obtained. In addition, the sharp Se∗-radii for functions belonging to several interesting classes are also determined.

In this work, the author reviews the origination of normalized tails of the Maclaurin power series expansions of infinitely differentiable functions, presents that the ratio between two normalized tails of the Maclaurin power series expansion of the exponential function is decreasing on the positive axis, and proves that the normalized tail of the Maclaurin power series expansion of the exponential function is absolutely monotonic on the whole real axis.

The main focus of this paper is to develop the solutions for the higher order difference equations with factorials and discrete exponential functions. Using these concept, we get the unique solution for the trigonometric exponential function for the initial valued problem. These results are verified using the numerical calculations.

In this paper, the authors prove global well-posedness of the massless Maxwell-Dirac equation in the Coulomb gauge on \\mathbb{R}^{1+d} (d\\geq 4) for data with small scale-critical Sobolev norm, as well as modified scattering of the solutions. Main components of the authors' proof are A) uncovering null structure of Maxwell-Dirac in the Coulomb gauge, and B) proving solvability of the underlying covariant Dirac equation. A key step for achieving both is to exploit (and justify) a deep analogy between Maxwell-Dirac and Maxwell-Klein-Gordon (for which an analogous result was proved earlier by Krieger-Sterbenz-Tataru, which says that the most difficult part of Maxwell-Dirac takes essentially the same form as Maxwell-Klein-Gordon.

This paper develops a generalized formulation of inverse difference operators for fractional extorial and exponential functions using the Gamma function. The fractional extorial function is expressed as a special case of the Mittag-Leffler factorial function, while fractional exponential functions are evaluated through Riemann-Liouville fractional integrals and derivatives, yielding expressions equivalent to the Mittag-Leffler function. A generalized representation is established and validated through lemmas, proofs, and numerical examples, thereby providing a unified analytical framework for fractional discrete and continuous operators.

This paper focuses on the generalized Calogero–Bogoyavlenskii–Schiff equation to extract new complex solutions by using two analytical methods, namely, Bernoulli sub-equation function method, Modified exponential function method. For better understanding of physical meanings of solutions, simulations are reported by using a computational package program. Moreover, strain conditions for validity of complex solutions are also archived. Finally, a conclusion part completes the paper by mentioning the novelties of paper.

In the present paper, we aimed to discuss certain coefficient-related problems for the inverse functions associated with a bounded turning functions class subordinated with the exponential function. We calculated the bounds of some initial coefficients, the Fekete–Szegö-type inequality, and the estimation of Hankel determinants of second and third order. All of these bounds were proven to be sharp.

We establish the linear independence of values of the q-analogue of the exponential function and its derivatives at specified algebraic arguments, when q is a Pisot–Vijayaraghavan number. We also deduce similar results for cognate functions, such as the Tschakaloff function and certain generalised q-series.

The extension of the exponential function is the Mittag-Leffler function. In physical implications, the MittagLeffler function is becoming more and more crucial, as a result a number of researchers are investigating different generalizations and extensions associated with the Mittag-Leffler function. In this present article, we define the bicomplex rparameter Mittag-Leffler function which is an extension of bicomplex one parameter Mittag-Leffler function. In addition, various properties of this function like integral representation, differential relation, duplication formula, bicomplex C-R equation, analyticity, region of convergence, order and type are established. Additionally, the bicomplex r-parameter MittagLeffler function's Laplace transform and Caputo fractional derivative are produced.