Document Type : Original Article

Authors

• Yogesh J. Bagul ¹
• Christophe Chesneau ²
• Ramkrishna M. Dhaigude ³

¹ Department of Mathematics, K. K. M. College, Manwath, Dist: Parbhani, Maharashtra - 431505, India

² Department of Mathematics, University of Caen Normandie, 14000 Caen, France

³ Department of Mathematics, Government Vidarbha Institute of Science and Humanities, Amravati, Maharashtra - 444604, India

Abstract

In this paper, we establish algebraic bounds of the ratio type in nature for the natural exponential function $e^x$ involving two parameters, a and n, which become optimal as a tends to 0 or n tends to infinity. The proof is mainly based on Chebyshev's integral inequality and properties of the incomplete gamma function. Subsequently, we focus on the simple case obtained with n = 1, with comparisons to existing literature results. For the applications, we provide alternative proofs of inequalities involving ratio functions of trigonometric and hyperbolic functions. Graphics are given to illustrate the theory.

Keywords

• Algebraic bounds
• optimal bounds
• exponential function
• ratio functions