SOME INTEGRAL MEAN INEQUALITIES CONCERNING POLAR DERIVATIVE OF A POLYNOMIAL

In this paper, we shall first obtain an integral inequality for the polar derivative of the above inequality. As an application of this result, we prove another inequality which is the Lr analogue of an inequality in polar derivative proved recently by Mir et al. [J. Interdisciplinary Math. 21(2018), 1387-1393].

Dubinin [6] used the Classical Schwarz Lemma and obtained an interesting refinement of (1.1) by proving that if P ∈ P n and P(z) has all it zeros in |z| ≤ 1, then For P ∈ P n , the polar derivative [9] of P(z) with respect to a point α, real or complex, is defined as Note that D α P(z) is polynomial of degree at most (n − 1). It generalizes the ordinary derivative in the sense that It is of interest to extend ordinary inequalities into polar derivatives because the later versions are the generalizations of the former.
Shah [13] extended inequality (1.1) to the polar derivative of P(z) and proved the following result.
Theorem 1.1. If P ∈ P n and P(z) has all its zeros in |z| ≤ 1, then for every complex number α Clearly Theorem 1.1 generalizes inequality (1.1) and to obtain (1.1) we simply divide both sides of (1.4) by |α| and let |α| → ∞.
Recently, Mir et al. [11] extended inequality (1.3) into its polar derivative version by proving: Theorem 1.2. If P ∈ P n and P(z) has all its zeros in |z| ≤ 1, then for every real or complex Inequality (1.5) is best possible and the extremal polynomial is p(z) = (z − 1) n with real α ≥ 1.
We know from analysis ( [12], [14]) that if P ∈ P n , then for each r > 0

MAIN RESULTS
In this paper, we extend inequality (1.3) to its integral analogue for the polar derivative of a polynomial and thereby obtain a generalization of it. Further, as an application of Theorem 2.1, we obtain a more general result which, as special cases, yield interesting generalizations and refinements of (1.2) and (1.3). First, we prove the following, which is the corresponding L r extension of Theorem 1.2.
Further, we prove the following theorem as an application of Theorem 2.1.
Theorem 2.3. If P ∈ P n and P(z) has all its zeros in |z| ≤ 1, then for every complex number α with |α| ≥ 1 and 0 ≤ t < 1, Taking limit as r → ∞ on both sides of (2.2) we have the following result concerning polar derivative recently proved by Mir et al. [11].
Corollary 2.5. If P ∈ P n and P(z) has all its zeros in |z| ≤ 1, then for every complex number α with |α| ≥ 1 and 0 ≤ t < 1,  Corollary 2.8. If P ∈ P n and P(z) has all its zeros in |z| ≤ 1, then for 0 ≤ t < 1, Remark 2.9. Taking limit as t → 1 in inequality (2.4) and using (1.6) we obtain an improved bound of inequality (1.2).

LEMMAS
For the proof of the theorems, we need the following lemmas.
The first lemma is due to Malik [8].
By applying Lemma 3.1 to Q(z) = z n P 1 z , we immediately get the following result.
Lemma 3.2. If P ∈ P n and P(z) has all its zeros in |z| ≤ k, k ≤ 1, then for |z| = 1, where Q(z) is defined as in Lemma 3.1.
Lemma 3.3. If P ∈ P n and P(z) has all its zeros in |z| ≤ 1, then for each point z on |z| = 1 at which P(z) = 0,

PROOF OF THE THEOREMS
Proof of Theorem 2.1. If Q(z) = z n P 1 z , it can be easily verified that for |z| = 1, Since P(z) has all its zeros in |z| ≤ 1, therefore, by Lemma 3.2 for k = 1, we have Now for every complex number α with |α| ≥ 1, we have for |z| = 1 which gives with the help of (4.1) For any r > 0 and 0 ≤ θ < 2π, from (4.2) we have which equivalently gives |c n | − |c 0 | |c n | + |c 0 | , which implies by using the fact Further, it is evident that inequality (4.4) follows trivially for those z on |z| = 1 at which P(z) = 0 as well.
Also from (4.4), we have for 0 ≤ θ < 2π and r > 0 This completes the proof of Theorem 2.1.
Since, all the zeros of P(z) lie in |z| < 1, it follows by Rouche's Theorem that all zeros of P(z) − λ mz n also lie in |z| < 1. Hence, by Theorem 2.1, we have for |α| ≥ 1 and for any r > 0, Since, for every λ with |λ | < 1, we have |c n − λ m| ≥ |c n | − m|λ |. and because the function (4.9) x − |c 0 | x + |c 0 | is a non-decreasing function of x, we have Also by triangle inequality, we have for |z| = 1, .7)]. (4.10) Applying the argument of (4.9) to the second factor and inequality (4.10) to the third factor of (4.8) respectively, we have Put |λ | = t in inequality (4.14), we get where 0 ≤ t < 1 and this completes the proof of Theorem 2.3.

CONFLICT OF INTERESTS
The author(s) declare that there is no conflict of interests.