Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Gateaux differentiability in the Banach space of meromorphic functions

Sanjay Mallick, Debmalya Sain

Abstract

We study the Gateaux differentiability in the Banach space of meromorphic functions and obtain a complete characterization of the same, by using Birkhoff-James orthogonality techniques. We introduce the concept of extended orthogonality covering set (EOCS), which allows us to present refinements of some earlier results on the Gateaux differentiability of analytic functions. We also discuss some related properties of meromorphic functions which follow directly from the said characterization.

Gateaux differentiability in the Banach space of meromorphic functions

Abstract

We study the Gateaux differentiability in the Banach space of meromorphic functions and obtain a complete characterization of the same, by using Birkhoff-James orthogonality techniques. We introduce the concept of extended orthogonality covering set (EOCS), which allows us to present refinements of some earlier results on the Gateaux differentiability of analytic functions. We also discuss some related properties of meromorphic functions which follow directly from the said characterization.

Paper Structure

This paper contains 4 sections, 7 theorems, 55 equations.

Table of Contents

  1. Introduction
  2. Main Results
  3. Statements and Declarations
  4. Acknowledgement

Key Result

Theorem 2.1

Let $f, g\in \mathcal{M}(D)$. Then the following are equivalent: $(i)$$f\perp_B g.$$(ii)$$\{ \left((f_{_Q}(z), g_{_Q}(z)), (f_{_R}(w), g_{_R}(w))\right) : (z,w)\in M_{f_{_Q}}\times M_{f_{_R}} \}$ is an EOCS.

Theorems & Definitions (24)

  • Definition 1.1
  • Definition 1.2
  • Theorem 2.1
  • proof
  • Remark 2.2
  • Theorem 2.3
  • Theorem 2.4
  • proof
  • Remark 2.5
  • Theorem 2.6
  • ...and 14 more