If P(z) is a polynomial of degree n, and α a complex number, then polar derivative of P(z) with respect to the point α, denoted by D α P(z), is defined by 

$$\displaystyle{D_{\alpha }P(z) = nP(z) + (\alpha -z)P^{{\prime}}(z).}$$

Clearly, D α P(z) is a polynomial of degree n − 1, and it generalizes the ordinary derivative in the sense that 

$$\displaystyle{\lim _{\alpha \rightarrow \infty }\left [\frac{D_{\alpha }P(z)} {\alpha } \right ] = P^{{\prime}}(z).}$$

It is well known that if P(z) is a polynomial of degree n, then max | z | = 1 | P ′ (z) | ≤ nmax | z | = 1 | P(z) | . This inequality is known as Bernstein’s inequality (Bernstein, Lecons sur les proprietes extremales et la meilleure approximation des fonctions analytiques d’une variable reelle. Gauthier-Villars, Paris, 1926), although this inequality was also proved by Riesz (Jahresber Dtsch Math-Verein 23:354–368, 1914) about 12 years before it was proved by Bernstein. The subject of inequalities for polynomials and related classes of functions plays an important and crucial role in obtaining inverse theorems in Approximation Theory. Frequently, the further progress in inverse theorems has depended upon first obtaining the corresponding analogue or generalization of Markov’s and Bernstein’s inequalities. These inequalities have been the starting point of a considerable literature in Mathematics, and was one of the areas in which Professor Q. I. Rahman worked for more than 50 years, and made some of the most important and significant contributions. Over a period, this Bernstein’s inequality and corresponding inequality concerning the growth of polynomials have been generalized in different domains, in different norms, and for different classes of functions, and in the literature one can find hundreds of papers on this topic. Here we study some of the research centered around Bernstein-type inequalities for polar derivatives of polynomials. The chapter is purely expository in nature and an attempt has been made here to provide results starting from the beginning to some of the most recent ones.

On Bernstein-Type Inequalities for the Polar Derivative of a Polynomial

In [S. Kumar and R. Lal, Generalizations of some polynomial inequalities, Int. Electron. J. Pure Appl. Math., 3, 2 (2011), 111–117.], Kumar and Lal provided an upper bound of a derivative for polynomial degree n having some of zeros at the origin and rest of zeros lying on or outside the boundary of a prescribed disk. In this paper, we present an upper bound of a derivative for polynomials p(z) = (z−zm)m(z−zm−1)m−1 · · · (z−z0)0 ( a0 + n−(tm+···+t0) ∑ ν=μ aν z ν ) of degree n having zeros z0, . . . ,zm with |z j |< 1 for 0 j m and the remaining n−(tm + · · ·+t0) zeros are outside {z : |z| < k} where k 1.

/pdf/an-upper-bound-of-a-derivative-for-some-class-of-polynomials-2dxd1n6y67.pdf

An Upper bound of a derivative for some class of polynomials

Certain problems concerning polynomials and transcendental entire functions of exponential type

If P(z) is a polynomial of degree n and α is any complex number, thenDαP(z)=nP(z)+(α−z)P′(z) is called the polar derivative of P(z) with respect to α, denoted by DαP(z). Chapter 6 is devoted to Bernstein-type inequalities for polar derivatives of polynomials. Beside extensions of the classical theorems of Grace and Laguerre, in this chapter are investigated the bounds on the uniform norm, as well as on the integral mean values, of polar derivative of polynomials. Also studied in this chapter are related inequalities for polynomials without restriction on their zeros, for polynomials having no zeros in a circle, as well as ones having all their zeros in a circle. 

Bernstein-type inequalities for polar derivatives of polynomials

IfP (z) = anz n + n P v= a n vz n v , 1 n, has all its zeros onjzj = k, k 1, then it was recently proved by Dewan and Ahuja (3) that for every real or complex number withj j k. maxzj=1 jD P (z)j n(j j +s )s k n +1 (1 +s ) maxzj=1 jP (z)j; In this paper, we improve the above result and obtain new inequality for the polar derivative of a polynomial.

/pdf/inequalities-for-the-polar-derivative-of-a-polynomial-28gqiorktv.pdf

Inequalities for the Polar Derivative of A Polynomial

If p(z) is a polynomial of degree n having all its zeros on |z| = k,k ≤ 1, then it is proved[5] that 

$$\mathop {\max }\limits_{\left| z \right| = 1} \left| {p'(z)} \right| \leqslant \frac{n} {{k^{n - 1} + k^n }}\mathop {\max }\limits_{\left| z \right| = 1} \left| {p(z)} \right|.$$

. In this paper, we generalize the above inequality by extending it to the polar derivative of a polynomial of the type $p(z) = c_n z^n + \sum\limits_{j = \mu }^n {c_{n - j} z^{n - j} } ,1 \leqslant \mu \leqslant n$. We also obtain certain new inequalities concerning the maximum modulus of a polynomial with restricted zeros.

On extremal properties for the polar derivative of polynomials

If is a polynomial of degree n having all its zeros in |z|≤ k , k ≥ 1 , then Govil [Proc Amer Math Soc, Vol 41 (1973), pp 543–546] proved In this paper, we generalize and extend the above inequality and some other related inequalities by restricting ourselves to polynomials of degree n having all its zeros in |z|≤ · k , k ≥ 1 with s -fold zeros at the origin where 0 ≤ s ≤ n 

Turán-type inequalities for polynomials with restricted zeros

In this paper we obtain certain results for the polar derivative of a polynomial $p(z) = c_nz^n +\sum_{j=\mu}^n c_{n-j}z^{n-j}$, $1\leq\mu\leq n$, having all its zeros on $|z| = k$, $k\leq 1$, which generalizes the results due to Dewan and Mir, Dewan and Hans We also obtain certain new inequalities concerning the maximum modulus of a polynomial with restricted zeros [Editor's note: There are flaws in the paper, see M A Qazi, Remarks on some recent results about polynomials with restricted zeros, Ann Univ Mariae Curie-Sklodowska Sect A 67 (2), (2013), 59-64 ]

/pdf/inequalities-concerning-polar-derivative-of-polynomials-4gcjvw65jb.pdf

Inequalities concerning polar derivative of polynomials

If is a polynomial of degree n, having all its zeros in |z| ≤ k, k ≤ 1, then it was shown by Govil [Proc. Amer. Math. Soc., 41 (2) (1973), 543–546] that In this paper we generalize as well as improve the above inequality and some other related inequalities by extending them to the polar derivative of a polynomial of degree n≥ 3having all its zeros in |z| ≥ k, k ≤ 1 with s-fold zeros at the origin, where 0≤s≤n. AMS Mathematics Subject Classification (2000): 30A10, 30C10, 30C15.

Turán-Type inequalities for polynomials with restricted Zeros -II

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On extremal properties for the polar derivative of polynomials

Turán-type inequalities for polynomials with restricted zeros

Inequalities concerning polar derivative of polynomials

Turán-Type inequalities for polynomials with restricted Zeros -II