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It is of interest to know the sharp bounds of the Hankel determinant, Zalcman functionals, Fekete-Szeg$ \ddot{o} $ inequality as a part of coefficient problems for different classes of functions. Let $\mathcal{H}$ be the class of functions…

Let A_n be the class of functions f(z) which are analytic in the open unit disk U} with f(0)=0, f'(0)=1, f"(0)=f"'(0)=...=f^{(n)}=0 and f^{(n+1)}\neq0. Applying the results due to S. S. Miller (J. Math. Anal. Appl. 65(1978), 289-305), some…

Second-order polynomials generalize classical first-order ones in allowing for additional variables that range over functions rather than values. We are motivated by their applications in higher-order computational complexity theory,…

For a normalised analytic function f defined on the open unit disk in the complex plane, we determine several sufficient conditions for starlikeness in terms of the quotients Q_{ST}:=zf'(z)/f(z), Q_{CV}:=1+zf"(z)/f'(z) and the Schwarzian…

The z-zeros of the modified Bessel function of the third kind K_{nu}(z), also known as modified Hankel function or Macdonald function, are considered for arbitrary complex values of the order nu. Approximate expressions for the zeros,…

The paper is devoted to a comprehensive second-order study of a remarkable class of convex extended-real-valued functions that is highly important in many aspects of nonlinear and variational analysis, specifically those related to…

The fractional integrals and fractional derivatives problem is tackled by using the operator approach. The definition domain E of operators is causal functions.Many properties of fractional integrals are given. Fractional derivatives…

This paper studies derivatives with respect to the parameters of Srivastava triple hypergeometric functions HA, HB and HC. Using basic properties of the Gamma function and Pochhammer symbols, we obtain explicit formulas for first and higher…

A generalization of the Apery-like numbers, which is used to describe the special values $\zeta_Q(2)$ and $\zeta_Q(3)$ of the spectral zeta function for the non-commutative harmonic oscillator, are introduced and studied. In fact, we give a…

This paper establishes sharp bounds for the second and third-order Toeplitz determinants associated with starlike functions $f$ in the unit disk such that $f(z)-z$ has a zero of order $k+1$ at $z=0$. These bounds are further extended to…

We generalize the notion of Erd\H{o}s-Ginzburg-Ziv constants -- along the same lines we generalized in earlier work the notion of Davenport constants -- to a ``higher degree" and obtain various lower and upper bounds. These bounds are…

We consider three classes of functions defined using the class $\mathcal{P}$ of all analytic functions $p(z)=1+cz+\dotsb$ on the open unit disk having positive real part and study several radius problems for these classes. The first class…

This is essentially a translated (and explained) version of a peper Hecke published in 1930 where he shows, for a prime q, a relation between the class number h(-q) and the representation of PSL(2, Z / pZ) on the space of holomorphic…

The set E of functions f fulfilling some conditions is taken to be the definition domain of s-order integral operator J^s (iterative integral), first for any positive integer s and after for any positive s (fractional, transcendental {\pi}…

We consider alternative orders of summation for the conditionally convergent series defining the weight-2 Eisenstein series G2 and the Weierstrass p-function. The resulting sums differ from the standard ones by a residual term that can be…

In this paper, we introduce a new subclass of harmonic functions $f=s+\overline{t}$ in the open unit disk $U =\left \{ z\in C:\left \vert z\right \vert <1\right \} $ satisfying ${\text{Re}}\left[ \gamma s^{\prime }(z)+\delta zs^{\prime…

We use the rationality of the generalized $h^{th}$ convergent functions, $Conv_h(\alpha, R; z)$, to the infinite J-fraction expansions enumerating the generalized factorial product sequences, $p_n(\alpha, R) =…

We establish the following results on higher order $\mathcal{S}^p$-differentiability, $1<p<\infty$, of the operator function arising from a continuous scalar function $f$ and self-adjoint operators defined on a fixed separable Hilbert…

Let $\mathcal{A}$ be the family of functions $f(z)=z+a_2z^2+...$ which are analytic in the open unit disc $\mathbb{D}=\{z: |z|<1 \}$, and denote by $\pe$ of functions $p(z)=z+p_1z+p_2z^2+...$ analytic in $\de$ such that $p(z)$ is in $\pe$…

Trigonometric formulas are derived for certain families of associated Legendre functions of fractional degree and order, for use in approximation theory. These functions are algebraic, and when viewed as Gauss hypergeometric functions,…