GENERALIZED Lᵖ INEQUALITIES FOR THE POLAR DERIVATIVE OF POLYNOMIAL 


By

Roshan Lal Keshtwal¹ , Susheel Kumar² and Imran Ali³

¹Department of Mathematics, V. S .K. C. Government Postgraduate College, Dakpathar, Dehradun, Uttarakhand, India-248125 

²Department of Mathematics, Deshbandhu College (University of Delhi), Kalkaji, New Delhi, India-110019 

³Department of Mathematics, Government Postgraduate College, Gopeshwar, Chamoli, Uttarakhand, India-246401 

Email: rlkeshtwal@gmail.com, skahlawatt@gmail.com, ali.imranmath@gmail.com 

(Received: February 13, 2023; In format: January 31, 2024; Revised: February 26, 2024 ; Accepted: March 12, 2024) 


DOI: https://doi.org/10.58250/jnanabha.2024.54110 


 

Abstract

Let P(z) be a polynomial of degree n and let α be any real or complex number, then the emanant or polar derivative of p(z) is denoted by Dαp(z) and defined as Dαp(z) = np(z) + (α − z)p 0 (z). The polynomial Dαp(z) is of degree at most n − 1 and it generalises the ordinary derivative p 0 (z) of p(z) in the sense that limα→∞ Dαp(z) α = p 0 (z). In this paper, we prove some L p inequalities for the emanant of the polynomial having all its zeros in prescribed disk. Our results generalize the earlier known results. 


2020 Mathematical Sciences Classification: 30D15, 30A10 

Keywords and Phrases: Polynomials; Derivatives; Polar Derivative; Integral Inequalities; Maximum Modulus; Emanant. 

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