A NOTE ON INEQUALITIES AND BOUNDS FOR LAMBERT W FUNCTION, AND ITS RELATIONSHIPS WITH GAMMA AND PSI FUNCTIONS
By
Mohammad Shakil¹ , Tassaddaq Hussain², R. C. Singh Chandel³ , Jai Narain Singh⁴,
Mohammad Ahsanullah⁵ and B. M. Golam Kibria⁶
1Department of Mathematics, Miami Dade College, Hialeah, FL, USA - 33012 ²Department of Statistics, Mirpur University of Science and Technology, Mirpur, Pakistan - 10250
3Department of Mathematics, D.V. Postgraduate College, Orai, Uttar Pradesh, India - 285001 4Department of Mathematics & Computer Science, Barry University, Miami Shores, FL, USA - 33161 5Professor Emeritus, Rider University, Lawrenceville, NJ, USA - 08648 6Department of Mathematics and Statistics, Florida International University, Miami, FL, USA - 33199
Email: mshakil@mdc.edu, tafkho2000@gmail.com, rc chandel@yahoo.com, jsingh@mail.barry.edu, ahsan@rider.edu, kibriag@fiu.edu
(Received: September 21, 2025; Revised: March 26, 2026; Accepted March 28, 2026)
DOI: https://doi.org/10.58250/jnanabha.2026.56101
Abstract
The Lambert W function, also known as the product logarithm, is an important special function with significant applications in mathematics, probability, combinatorics, and physics. This paper investigates inequalities and bounds for the Lambert W function and explores its relationships with the Gamma and Psi functions. Derivations, approximations, and bounds are provided. Figures demonstrate the accuracy of bounds and approximations.
2020 Mathematical Sciences Classification: 26D07, 33B15, 33E50, 68Q17.
Keywords and Phrases: Asymptotics; Bounds; Gamma function; Inequalities; Lambert W function; Psi (digamma) function; Special functions.