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Wilker and Huygens type inequalities for mixed trigonometric-hyperbolic functions

Yogesh J. Bagul, Ramkrishna M. Dhaigude, Barkat A. Bhayo, Vinay M. Raut

Abstract

Motivated by the work of J. Sándor [19], in this paper we establish a new Wilker type and Huygens type inequalities involving the trigonometric and hyperbolic functions. Moreover, in terms of hyperbolic functions, the upper and lower bounds of sin(x)/x and tan(x)/x are given.

Wilker and Huygens type inequalities for mixed trigonometric-hyperbolic functions

Abstract

Motivated by the work of J. Sándor [19], in this paper we establish a new Wilker type and Huygens type inequalities involving the trigonometric and hyperbolic functions. Moreover, in terms of hyperbolic functions, the upper and lower bounds of sin(x)/x and tan(x)/x are given.

Paper Structure

This paper contains 3 sections, 13 theorems, 74 equations.

Table of Contents

  1. Introduction
  2. Preliminaries and Lemmas
  3. Proofs and corollaries

Key Result

theorem 1

For $0 < x < \pi/2$ it is asserted that

Theorems & Definitions (21)

  • theorem 1
  • theorem 2
  • theorem 3
  • theorem 4
  • lemma 1
  • lemma 2
  • lemma 3
  • proof
  • lemma 4
  • lemma 5
  • ...and 11 more