ON MAXIMUM MODULUS OF POLAR DERIVATIVE OF A POLYNOMIAL

In this paper, we consider the more general class of polynomials p(z) = a0+ n ∑ ν=μ aν zν , 1≤ μ ≤ n, not vanishing in |z| < k, k > 0, to estimate max |z|=ρ |Dα p(z)| in terms of max |z|=r |p(z)| by involving some coefficients of p(z), where 0 < r ≤ ρ ≤ k. Interestingly, the results improve and extend other well known inequalities to polar derivative. Moreover, our results give several interesting results as special cases.


INTRODUCTION AND STATEMENT OF RESULTS
Let p(z) be a polynomial of degree n and let M(p,t) = max |z|=t |p(z)|, and m(p,t) = min |z|=t |p(z)|.
Concerning the estimate of max |p (z)| in terms of max |p(z)| on the unit circle |z| = 1, we have M(p , 1) ≤ nM(p, 1) . (1.1) Inequality (1.1) is the famous result known as Bernstein's inequality [17] and equality holds if and only if p(z) has all its zeros at the origin.
As a generalization of inequality (1.2), Malik [14] proved that if p(z) is a polynomial of degree n not vanishing in |z| < k, k ≥ 1, then The result is best possible and equality holds for p(z) = (z + k) n .

) by proving
Theorem A. If p(z) = n ∑ ν=0 a ν z ν is a polynomial of degree n having no zero in |z| < k, k ≥ 1, then for 0 < r ≤ ρ ≤ k, M(p , ρ) ≤ n(k + ρ) n−1 (k + r) n 1 − k(k − ρ)(n|a 0 | − k|a 1 |)n (k 2 + ρ 2 )n|a 0 | + 2k 2 ρ|a 1 | × ρ − r k + ρ k + r k + ρ Further, Mir et al. [15] recently generalized and improved Theorem A by establishing Theorem B. If p(z) = a 0 + n ∑ ν=µ a ν z ν , 1 ≤ µ ≤ n, is a polynomial of degree n having no zero in |z| < k, k > 0, then for 0 < r ≤ ρ ≤ k, where here and throughout this paper, Let D α p(z) denotes the polar derivative of a polynomial p(z) of degree n with respect to a real or complex number α, then The polynomial D α p(z) is of degree at most n − 1 and it generalizes the ordinary derivative in the sense that Aziz [1] extended Theorem A to the polar derivative of p(z) by proving a ν z ν is a polynomial of degree n having no zero in |z| < k, k ≥ 1, then for every real or complex number α, with |α| ≥ 1, Inequality (1.7) is best possible for p(z) = (z + k) n with a real number α ≥ 1 and k ≥ 1.
In this paper, we consider the more general class of polynomials p(z) = a 0 + n ∑ ν=µ a ν z ν , 1 ≤ µ ≤ n, not vanishing in |z| < k, k > 0, with 0 < r ≤ ρ ≤ k, and find estimates of the maximum modulus of the polar derivative of p(z) on the circle |z| = ρ in terms of M(p, r) by involving some of the coefficients of p(z). In fact, we first prove Theorem 1.1. If p(z) = a 0 + n ∑ ν=µ a ν z ν , 1 ≤ µ ≤ n, is a polynomial of degree n having no zero in |z| < k, k > 0, then for every real or complex number α with |α| ≥ ρ and 0 < r ≤ ρ ≤ k, For µ = 1, the result is best possible with α > 0 and the extremal polynomial is p(z) = (z + k) n .
and hence under the same hypotheses of Theorem 1.1, we have the following result which gives a generalised extension of Theorem A to polar derivative.
For µ = 1, and r = ρ = 1, Theorem 1.1 reduces to Theorem C. Remark 1.3. If we divide both sides of (1.8) by |α| and make |α| → ∞, we have the following interesting generalization as well as an improvement of Theorem A.
Remark 1.5. Dividing both sides of inequality (1.9) by |α| and letting |α| → ∞, we obtain the following generalization of Theorem A. Corollary 1.6. If p(z) = a 0 + n ∑ ν=µ a ν z ν , 1 ≤ µ ≤ n, is a polynomial of degree n having no zero in |z| < k, k > 0, and for 0 < r ≤ ρ ≤ k, This result is best possible and equality holds for p(z) = (z µ + k µ ) n µ , where n is a multiple of µ.
Next, under the same set of hypotheses of Theorem 1.1, it is of interest to find better bound than that of Theorem 1.1, by involving m(p, k) = min |z|=k |p(z)|. In this context, more precisely, we are able to prove the following significant result, which gives improved extension of Theorem B to polar derivative.
is a polynomial of degree n having no zero in |z| < k, k > 0, then for every real or complex number α with |α| ≥ ρ and 0 < r ≤ ρ ≤ k, For µ = 1, the result is best possible with α > 0 and the extremal polynomial is p(z) = (z + k) n . Remark 1.8. When r = ρ = 1, inequality (1.13) of Theorem 1.7 reduces to (2.18) of Lemma 2.10 due to Dewan and Singh [10].
Further, Theorem 1.7 gives an improvement of a result proved by Dewan and Singh [10, Theorem 2]. Moreover, when µ = 1, and r = ρ = 1, Theorem 1.7 reduces to a result of Aziz and Shah [3], which improves upon Theorem C.
Dividing both sides of (1.13) by |α| and letting |α| → ∞, we have the following interesting result, which improves upon Theorem B.
This result is best possible and equality occurs for p(z) = (z µ + k µ ) n µ , where n is a multiple of µ. Remark 1.10. As mentioned earlier, Corollary 1.9 gives a sharper bound than that of Theorem B, because of the following facts: This inequality is best possible for µ = 1 with α > 0 and the extremal polynomial being p(z) = (z + k) n , k > 0.
For µ = 1, Corollary 1.12 gives an improved extension to polar derivative of a result due to Aziz and Zargar [5]. The bound of Corollary 1.12 improves upon a result due to Dewan and Singh [10, Corollary 1]. Further for µ = 1, it gives an improvement of inequality (1.5) of Theorem A.

LEMMAS
The following lemmas are needed for the proofs of the theorems. Lemma 2.1. If p(z) = a 0 + n ∑ ν=µ a ν z ν , 1 ≤ µ ≤ n, is a polynomial of degree n having no zero in The above result is due to Chan and Malik [7].
We are interested to prove the following lemma concerning the validity of inequality (2.4) of Lemma 2.3 for any k > 0, because this has subsequent uses.
Lemma 2.6. If p(z) = a 0 + n ∑ ν=µ a ν z ν , 1 ≤ µ ≤ n, is a polynomial of degree n having no zero in |z| < k, k > 0, then the function is a non-decreasing function of t in (0, k].
Proof of Lemma 2.6. We prove this by derivative test. Now, we have which is non-negative, since by Lemma 2.5, (n|a 0 | − µ|a µ |k µ ) ≥ 0, and the fact that t ≤ k. Lemma 2.7. For x ∈ (0, 1] and any r ≥ 1, we have Proof of Lemma 2.7. For r = 1, the result follows trivially. Hence to prove Lemma 2.7, it is sufficient to show that for r > 1, Hence f (x) is a non-decreasing function of x in (0, 1] and therefore for x ∈ (0, 1], we have which is inequality (2.10) and the proof of the lemma is complete.
Lemma 2.8. If p(z) = a 0 + n ∑ ν=µ a ν z ν , 1 ≤ µ ≤ n, is a polynomial of degree n having no zero in |z| < k, k > 0, then for 0 < r ≤ ρ ≤ k, The result is best possible and inequality holds for p(z) = (z µ + k µ ) n µ , where n is a multiple of µ.

This gives
M(p,t)dt. (2.14) Using inequality (2.6) of Lemma 2.4 with ρ = t and noting that 0 (By Lemma 2.6, the factor in the integrand in (2.15) which is same as f (t) of Lemma 2.6, is a non-decreasing function of t in (0, k]) which proves Lemma 2.8.
Lemma 2.9. If p(z) = a 0 + n ∑ ν=µ a ν z ν , 1 ≤ µ ≤ n, is a polynomial of degree n having no zero in The result is best possible and equality holds for p(z) = (z µ + k µ ) n µ , where n is a multiple of µ.
Proof of Lemma 2.9. The proof of this lemma follows on the same lines as that is in Lemma 2.8, but instead of applying (2.6) of Lemma 2.4, we use (2.5) of the same lemma. We omit the details. These two results are due to Dewan and Singh [10].
Integrating g(t) with respect to t from r to ρ, we shall obtain the quantity U, which is necessarily non-negative.
Lemma 2.13. If µ, n are positive integers such that 1 ≤ µ ≤ n, then for 0 < r ≤ ρ ≤ k, we have where U is defined in Lemma 2.12.