On Turán and Bernstein-type integral mean estimates for polar derivative of a complex polynomial

Nirmal Kumar Singha ; Barc ; and Chanam

Publications de l'Institut Mathématique, _N_S_118 (2025) no. 132, p. 97

Résumé

If $p(z)$ is a polynomial of degree $n$ having all its zeros in $|z|\leq k$, $k\geq1$, then for $r\geq 1$, Aziz [J. Approx. Theory, extbf{55} (1988), 232--239] proved \[ \bigg\{\int_0^{2i}|1+k^ne^{iheta}|^rdheta\bigg\}^{1/r}\max_{|z|=1}|p'(z)|\geq n\bigg\{\int_0^{2i}|p(e^{iheta})|^rdheta\bigg\}^{1/r}, \] whereas Devi et al. [Note Mat., 41 (2021), 19--29] proved that if $p(z)$ is a polynomial of degree $n$ having no zero in $|z|$, then for $r>0$, \[ k^nn\bigg\{\int_0^{2i}|p(e^{iheta})|^rdheta\bigg\}^{1/r} eq\bigg\{\int_0^{2i}|e^{iheta}+k^n|^rdheta\bigg\}^{1/r}\Big\{n\max_{|z|=1}|p(z)|-\max_{|z|=1}|p'(z)|\Big\}, \] provided $|p'(z)|$ and $|q'(z)|$ attain their maxima at the same point on $|z|=1$, where $q(z)=z^n\overline{p\left(1/\bar z\right)}$. We not only obtain improved extensions of the above inequalities into polar derivative by involving the leading coefficient and the constant term of the polynomial but also give integral analogues of inequalities on polar derivative recently proved by Mir and Dar [Filomat, extbf{36}(16) (2022), 5631--5640].

Détail

DOI : 10.2298/PIM2532107S

Classification : 30A10, 30C10, 30C15
Keywords: polar derivative, Turán-type inequalities, Bernstein-type inequalities, integral inequalities