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Finite differences have been widely used in mathematical theory as well as in scientific and engineering computations. These concepts are constantly mentioned in calculus. Most frequently-used difference formulas provide excellent…

Sharp estimates on $\beta$ are determined so that an analytic function $p$ defined on the open unit disk in the complex plane normalized by $p(0)=1$ is subordinate to some well known starlike functions with positive real part whenever…

The existence of Voiculescu's subordination functions in the context of non-tracial operator-valued C*-probability spaces has been established using analytic function theory methods. We use a matrix construction to show that the…

Using the notion of h-subdifferential, we characterize both first and second order differentiability of h-convex functions in stratified groups. We show that Aleksandrov's second order differentiability of h-convex functions is equivalent…

The purpose of this work is to show that the Khalil and Katagampoula conformable derivatives are equivalent to the simple change of variables $x$ $\rightarrow $ $x^{\alpha }/\alpha ,$ where $\alpha $ is the order of the derivative operator,…

A typical quandary in geometric functions theory is to study a functional composed of amalgamations of the coefficients of the pristine function. Conventionally, there is a parameter over which the extremal value of the functional is…

In the early 1960's, Moreau and Rockafellar introduced a concept of called \emph{subgradient} for convex functions, initiating the developments of theoretical and applied convex analysis. The needs of going beyond convexity motivated the…

This paper introduces a second-order differential inclusion for unconstrained convex optimization. In continuous level, solution existence in proper sense is obtained and exponential decay of a novel Lyapunov function along with the…

These notes focus on the minimization of convex functionals using first-order optimization methods, which are fundamental in many areas of applied mathematics and engineering. The primary goal of this document is to introduce and analyze…

We derive global analytic representations of fundamental solutions for a class of linear parabolic systems with full coupling of first order derivative terms where coefficient may depend on space and time. Pointwise convergence of the…

This article is centered around generalizing a previous implicit function theorem of the author to be applicable for maps f:E sqcap F to F which can be lifted to Keller C^k_pi maps f_i:E sqcap F_i to F_i with F_i Banach and F=projlim F_i .…

In the present paper, we define a new general subclass of bi-univalent functions involving a differential operator in the open unit disk U. For this purpose, we use the Faber polynomial expansions. Several connections to some of the earlier…

The completeness of the group classification of systems of two linear second-order ordinary differential equations with constant coefficients is delineated in the paper. The new cases extend what has been done in the literature. These cases…

We extend the standard notion of self-concordance to non-convex optimization and develop a family of second-order algorithms with global convergence guarantees. In particular, two function classes -- \textit{weakly self-concordant}…

Biquaternionic Vekua-type equations arising from the factorization of linear second order elliptic operators are studied. Some concepts from classical pseudoanalytic function theory are generalized onto the considered spatial case. The…

In this paper, our focus lies on the study of the second-order variational analysis of orthogonally invariant matrix functions. It is well-known that an orthogonally invariant matrix function is an extended-real-value function defined on…

In this paper, we introduce a new second-order directional derivative and a second-order subdifferential of Hadamard type for an arbitrary nondifferentiable function. We derive several second-order optimality conditions for a local and a…

For analytic functions f(z) in the open unit disk U with f(0)=f'(0)-1=0, S. S. Miller and P. T. Mocanu (Integral Transform. Spec. Funct. 19(2008)) have considered some sufficient problems for starlikeness. The object of the present paper is…

In the Euclidean setting the celebrated Aleksandrov-Busemann-Feller theorem states that convex functions are a.e. twice differentiable. In this paper we prove that a similar result holds in the Heisenberg group, by showing that every…

Properties of the mappings \begin{align*} C&\mapsto\frac1{(2\pi i)^2}\int_{\Gamma_1}\int_{\Gamma_2}f(\lambda,\mu)\,R_{1,\,\lambda}\,C\, R_{2,\,\mu}\,d\mu\,d\lambda, C&\mapsto\frac1{2\pi i}\int_{\Gamma}g(\lambda)R_{1,\,\lambda}\,C\,…