Inequalities concerning sth derivative of a polynomial

If p(z) is a polynomial of degree n having no zero in |z| < k, k > 1, then for 0 < s < n and 1 ≤ R ≤ k Jain [2007 Turk. J. Math. 31 89-94] proved max|z|=R|p(s)(z)|≤1Rs+ks[{dsdxs(1+xn)}x=1](R+k1+k)nmax|z|=1|p(z)| In this paper, we improve as well as extend this inequality by involving certain coefficients of the polynomial. Further, our result improves and generalizes some well-known inequalities.

In this paper, we improve as well as extend this inequality by involving certain coefficients of the polynomial. Further, our result improves and generalizes some well-known inequalities.

Introduction and Statement of Results
Let P n be the class of polynomials p(z) = n j=0 a j z j of degree n. For a polynomial p ∈ P n , we denote M (p, R) = max |z|=R |p(z)| and p (z) the derivative of p(z).
If we consider p ∈ P n such that p(z) = 0 inside the disk |z| < 1, Erdös conjectured that inequality (1.1) can be sharpened and replaced by Inequality (1.2) was later proved by Lax [9]. Several refinements and extensions of (1.2) have been added to literature over the years (see Malik [10], Bidkham and Dewan [3], Jain [8]). 3) and proved that if p ∈ P n and p(z) = 0 in |z| < k, k ≥ 1, then for 1 ≤ R ≤ k, Jain [8] further extended (1.4) by considering the s th derivative of the polynomial. In fact, he proved Theorem A. If p ∈ P n and p(z) = 0 in |z| < k, k ≥ 1, then for 0 ≤ s < n and 1 ≤ R ≤ k, The result is sharp and equality holds with s = 1 for p(z) = (z + k) n .
Theorem A was further generalized and improved by Barchand and Dewan [5] by proving Theorem B. If p ∈ P n and p(z) = 0 in |z| < k, k > 0, then for 0 < r ≤ R ≤ k, and 1 ≤ s < n, In this paper, by involving certain coefficients of the polynomial, we obtain an extension and improvement of (1.5) and an improvement of (1.6). More precisely, we prove Theorem. If p ∈ P n and p(z) = 0 in |z| < k, k > 0, then for 0 ≤ s < n, and for 0 < r ≤ R ≤ k, and where here and throughout the paper It follows by Rouche's theorem that, for any real or complex number λ which leads to (1.13) (1.14) It is evident from (1.14) of Remark 1 and Lemma 2.8 that our theorem is an improvement of Theorem B due to Barchand and Dewan [4].

Remark 2.
Putting r = 1 in the theorem, we have the following improvement of Theorem A. Corollary 1. If p ∈ P n and p(z) = 0 in |z| < k, k ≥ 1, then for 0 ≤ s < n, and for 1 ≤ R ≤ k, where (1.17) It is clear from inequality (1.14) of Remark 1 in conjunction with Lemma 2.8 for r = 1 that Corollary 1 is an improvement of Theorem A due to Jain [8].

Lemmas
We need the following lemmas to prove our result. The first lemma is due to Aziz and Rather [1].
(2.6) Remark 6. If we take µ = 1 in Lemma 2.3, then from (2.6), B becomes which, from the proof of Lemma 2.8 equals B given by (1.9) in the theorem. Thus, for µ = 1, we obtain from (2.5) where a s is any complex number and k ≥ 1, is non-increasing for all non-zero real x.
Proof of Lemma 2.4. The proof follows from the first derivative test of f (x) for any non-zero real x and k ≥ 1, that is, c(n, s)(1 + k s+1 ) + |a s | x (k s+1 + k 2s ) 2 ≤ 0.