        TY  - JOUR
T1  - On the polar derivative of lacunary type polynomials
AU  - Motamednezhad, Ahmad
AU  - Mohammadi, Fatemeh
PY  - 2024
DA  - December
Y2  - 2024
DO  - 10.31801/cfsuasmas.1521079
JF  - Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics
JO  - Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat.
PB  - Ankara University
WT  - DergiPark
SN  - 1303-5991
SP  - 1197
EP  - 1209
VL  - 73
IS  - 4
LA  - en
AB  - Let $p(z)=a_nz^n+\sum_{l=\nu}^na_{n-l}z^{n-l}$, where $1\leq \nu \leq n$,  be a polynomial of degree $n$ having all its zeros in $|z|\leq k\leq 1$.  For polar derivative $D_{\alpha}p(z)$, it is known that for each $|\alpha|\leq 1$ on $|z|=1$,\begin{align*}|D_{\alpha}p(z)|\leq  \frac{n}{1+k^{\nu}}\Big\{(|\alpha|+k^{\nu})\|p\|_{\infty}-\frac{1-|\alpha|}{k^{n-\nu}}\min_{|z|=k}|p(z)|\Big\}.\end{align*}In this paper, we obtain the $L_q$ mean extension  and a refinement of the above and other related results for the polar derivative of polynomials.
KW  - Derivative
KW  - lacunary type polynomial
KW  - $L^q$ Inequality
KW  - maximum modulus
KW  - restricted zeros
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UR  - https://doi.org/10.31801/cfsuasmas.1521079
L1  - https://dergipark.org.tr/en/download/article-file/4091937
ER  -