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Classical exponential bounds for p-generalized circular and hyperbolic functions

Yogesh J. Bagul, Bharti O. Fande

Abstract

In this article, we obtain exponential bounds for the generalized circular and hyperbolic functions with one parameter p. Our results are natural generalizations of some existing results for classical circular and hyperbolic functions.

Classical exponential bounds for p-generalized circular and hyperbolic functions

Abstract

In this article, we obtain exponential bounds for the generalized circular and hyperbolic functions with one parameter p. Our results are natural generalizations of some existing results for classical circular and hyperbolic functions.
Paper Structure (3 sections, 10 theorems, 60 equations)

This paper contains 3 sections, 10 theorems, 60 equations.

Table of Contents

  1. Introduction
  2. Preliminaries and Lemmas
  3. Main results

Key Result

lemma 1

(anderson) Let $f_1(x)$ and $f_2(x)$ be two real valued-functions which are continuous on $[a, b]$ and derivable on $(a, b)$, where $-\infty < a < b < \infty$ and $f _2^{\prime}(x) \neq 0,$ for all $x \in (a, b).$ Let, and Then, we have The strictness of the monotonicity of $A(x)$ and $B(x)$ depends on the strictness of the monotonicity of $f_1^{\prime}(x)/f_2^{\prime}(x)$.

Theorems & Definitions (18)

  • lemma 1
  • lemma 2
  • proof
  • lemma 3
  • proof
  • theorem 1
  • proof
  • theorem 2
  • proof
  • theorem 3
  • ...and 8 more