• Nonlinear Functional Analysis and Applications
• /
• v.28 no.1
• /
• pp.287-294
• /
• 2023

The goal of this paper is to extend some inequalities of Bernstein as well as Turán type to polar derivative of a polynomial.

• Nonlinear Functional Analysis and Applications
• /
• v.28 no.3
• /
• pp.731-751
• /
• 2023

If p(z) is a polynomial of degree n having all its zeros in |z| ≤ k, k ≥ 1, then for any complex number α with |α| ≥ k, and r ≥ 1, Aziz [1] proved $$\left{{\int}_{0}^{2{\pi}}\,{\left|1+k^ne^{i{\theta}}\right|^r}\,d{\theta}\right}^{\frac{1}{r}}\;{\max\limits_{{\mid}z{\mid}=1}}\,{\mid}p^{\prime}(z){\mid}\,{\geq}\,n\,\left{{\int}_{0}^{2{\pi}}\,{\left|p(e^{i{\theta}})\right|^r\,d{\theta}\right}^{\frac{1}{r}}.$$ In this paper, we obtain an improved extension of the above inequality into polar derivative. Further, we also extend an inequality on polar derivative recently proved by Rather et al. [20] into L^r-norm. Our results not only extend some known polynomial inequalities, but also reduce to some interesting results as particular cases.

• Korean Journal of Mathematics
• /
• v.30 no.2
• /
• pp.199-203
• /
• 2022

Let f(z) be an entire function of exponential type τ such that ║f║ = 1. Also suppose, in addition, that f(z) ≠ 0 for ℑz > 0 and that $h_f(\frac{\pi}{2})=0$. Then, it was proved by Gardner and Govil [Proc. Amer. Math. Soc., 123(1995), 2757-2761] that for y = ℑz ≤ 0 $${\parallel}D_{\zeta}[f]{\parallel}{\leq}\frac{\tau}{2}({\mid}{\zeta}{\mid}+1)$$, where D_ζ[f] is referred to as polar derivative of entire function f(z) with respect to ζ. In this paper, we prove an inequality in the opposite direction and thereby obtain some known inequalities concerning polynomials and entire functions of exponential type.