Initial Coefficients and Fekete-Szegő Inequalities for Functions Related to van der Pol Numbers (VPN)

Abstract

The purpose of this paper is to find coefficient estimates for the class of functions ℳ N ( γ , ϑ , λ ) consisting of analytic functions f normalized by f(0) = f′(0) – 1 = 0 in the open unit disk D subordinated to a function generated using the van der Pol numbers, and to derive certain coefficient estimates for a 2, a 3, and the Fekete-Szegő functional upper bound for f ∈ ℳ N ( γ , ϑ , λ ) . Similar results were obtained for the logarithmic coefficients of these functions. Further application of our results to certain functions defined by convolution products with a normalized analytic functions is given, and in particular, we obtain Fekete-Szegő inequalities for certain subclasses of functions defined through the Poisson distribution series.

Ozaki-Type Bi-Close-to-Convex and Bi-Concave Functions Involving a Modified Caputo’s Fractional Operator Linked with a Three-Leaf Function

“…We conclude that one can determine new results for various new subclasses of OCV Σ (℘; ϑ) and BCV ϑ τ (λ) by selecting a specific Λ related to the series associated with telephone numbers, Van der Pol Numbers (VPNs), Gregory coefficients, and rational functions (see [11,12,48]). In conclusion, considering the tremendous amount of research embedding q-calculus into geometric function theory (see, for example, [49] and the references stated therein), this research could inspire future q-calculus developments for the above-mentioned classes by applying some obvious parametric and argument variations.…”