On the generalized derivative of a polynomial with restricted zeros

Let \(p(z)=c\prod _{j=1}^{n}(z-z_j)\) $p(z)=c{\prod }_{j=1}^n(z-z_j)$ with \(~c\ne 0\) $c\ne 0$ be a polynomial of degree n and \(p^\gamma (z)\) $p^{\gamma }(z)$ be its generalized derivative, where \(0\ne \gamma =(\gamma _1,\gamma _2,\ldots ,\gamma _n)\) $0\ne \gamma =({\gamma }_1,{\gamma }_2,\dots ,{\gamma }_n)$ is an n-tuples of non-negative real numbers in the Euclidean space \(\mathbb {R}^n\) ${\mathbb{R}}^n$ . In this paper, we consider a class of polynomials having all zeros in \(|z|\le \rho , ~\rho \le 1\) $|z|\le \rho ,\rho \le 1$ and investigate the dependence of \(\max _{|z|=1}\bigg |zp^\gamma (z)+\frac{\Lambda \beta }{1+\rho }p(z)\bigg |\) ${max}_{|z|=1}|z{p}^{\gamma }(z)+\frac{\mathrm{\Lambda }\beta }{1+\rho }p(z)|$ on \(\max _{|z|=1} |p(z)|,\) ${max}_{|z|=1}|p(z)|,$ where \(\beta \) $\beta$ is any complex number with \(|\beta |\le 1\) $|\beta |\le 1$ and \(\Lambda =\sum _{j=1}^{n}\gamma _j.\) $\mathrm{\Lambda }={\sum }_{j=1}^n{\gamma }_j.$ Our results not only generalize several established polynomial inequalities, but also enable the derivation of a variety of interesting outcomes through a consistent approach. In addition to it, we also present some generalizations of Turán’s inequality.