Generalized and Extended Versions of Ankeny–Rivlin and Improved, Generalized, and Extended Versions of Rivlin Type Inequalities for the s th Derivative of a Polynomial

: Let p ( z ) be a polynomial of degree n having no zeros in | z | < 1. In this paper, we generalize and extend a well-known result proven by Ankeny and Rivlin for the s th derivative of the polynomial. Furthermore, another well-known result proven by Rivlin is also improved, generalized and extended for the s th derivative of p ( z ) . Our results also give a number of interesting consequences as special cases.


Introduction
Let p(z) be a polynomial of degree n with real or complex coefficients. We denote M(p, r) = max |z|=r |p(z)| and m(p, r) = min |z|=r |p(z)|.
If p(z) is a polynomial of degree n, then it is well-known that: M(p , 1) ≤ n M(p, 1). (1) The result is best possible and the equality holds in (1) for p(z) = αz n , α being a complex number. This inequality, which is known as Bernstein's inequality, was proven by Bernstein [1], although it was also proven by Riesz [2] about 12 years before Bernstein.
Erdös considered the class of polynomials p(z) of degree n having no zeros in |z| < 1 and conjectured that: The result is sharp and the equality in (2) holds for p(z) = αz n + β, where α and β are complex numbers such that |α| = |β|. This inequality (2) was later verified by Lax [3] in 1944.
It is also well-known that if p(z) is a polynomial of degree n, then: M(p, R) ≤ R n M(p, 1), R ≥ 1 (3) and M(p, r) ≥ r n M(p, 1), r ≤ 1.
Inequality (3) is a consequence of the Maximum Modulus Principle [4] (p. 137 Problem III 269). Equality in (3) holds if p(z) is a constant multiple of z n . Whereas, inequality (4) is due to Zarantonello and Varga [5].
While for the same class of polynomials, Rivlin [7] improved (4) by proving the following result. Theorem 2. If p(z) is a polynomial of degree n having no zeros in |z| < 1, then: M(p, r) ≥ r + 1 2 n M(p, 1), r ≤ 1.
There are many extensions of inequality (5) (see Govil [8], Qazi [9], Bidkham and Dewan [10], Govil and Qazi [11], Govil et al. [12], Mir [13]). Some of the results by the above authors considered a more general class of polynomials while some also involve certain coefficients in the inequalities.
Jain [14] proved a generalization of Theorem 1 by simultaneously considering polynomials not vanishing in |z| < k, k ≥ 1 as well as the s th derivative of the polynomials. In fact, he proved: Theorem 4. If p(z) is a polynomial of degree n having no zeros in |z| < k, k ≥ 1, then for 0 ≤ s < n, and Equality holds in (8) with k = 1 and s = 0 for p(z) = z n + 1 and in (9) for s = 1 and p(z) = (z + k) n .
The paper is organized as follows. Section 2 includes preliminary results in the form of lemmas which are required to prove the main results and other claims. In Section 3, we state the main theorems of the paper and their implications on the existing results with the help of corollaries and remarks. Section 4 gives the proofs of the main theorems and in Section 5, a conclusion with some possible future prospects of the results is given.

Lemmas
The following lemmas are required to prove the results which are stated in the next section.
a ν z ν is a polynomial of degree n having no zeros in |z| < k, k ≥ 1, then: furthermore: where, The above lemma is due to Govil et al. [15]. In their paper [15], the authors mentioned that the bound given by (11) improves over the bound given by (10) because of the fact: However, the authors did not discuss it. For the sake of completeness, we give a brief proof of it in Lemma 3 by applying the next lemma. Lemma 2. Let a, b, c, d > 0 be real numbers such that c ≤ d. If a ≤ b then: Proof of Lemma 2.2. Suppose that c < d. Then inequality (14) follows easily as: which simplifies to: which gives the desired conclusion of the lemma. For the case c = d equality in (14) holds trivially.
We now show that the bound given by (11) is smaller than the bound given by (10).
a ν z ν be a polynomial of degree n having no zeros in |z| < k, k ≥ 1, then: where λ and µ are given by (12).
Now by (13), c ≤ d and it is easy to see that a ≤ b. Hence, it follows from inequality (14) of Lemma 2 that: Equivalently, which is the desired conclusion of the lemma. a ν z ν be a polynomial of degree n having no zeros in |z| < k, k > 0, where λ and µ are given by (12).

Lemma 5.
Let P(z) be a polynomial of degree n having all its zeros in |z| ≤ 1. If p(z) is a polynomial of degree at most n such that: |p(z)| ≤ |P(z)| on |z| = 1, Lemma 6. If p(z) is a polynomial of degree at most n, then for 0 ≤ s < n, where, Lemma 5 and Lemma 6 are due to Jain [14].
a ν z ν is a polynomial of degree n having no zeros in |z| < k, k > 0, then where, λ and µ are given by (12).
a ν z ν is a polynomial of degree n ≥ 3 having no zeros in |z| < k, k > 0, where E t,k is given by (21).
Proof of Lemma 8. If p(z) has a zero on |z| = k, then m(p, k) = 0 and because of Lemma 7, the result holds trivially. Hence, without loss of generality, we assume that p(z) has all its zeros in |z| > k, so that m > 0 and z n p(z) is analytic in |z| ≤ k. Then, by Maximum Modulus Principle [4] (p. 137, problem III 269), we have for |z| ≤ k: Hence, we have: m|z| n k n ≤ |p(z)| for |z| ≤ k.
The next lemma was proved by Aziz and Rather Corollary 1 in [16].
a ν z ν is a polynomial of degree n having no zeros in |z| < k, k ≥ 1, then and where C(n, s) = n! (n − s)! s! .
Remark 1. From Lemma 9, it is easy to conclude that for 0 ≤ s < n, and The following result is due to Jain Remark 1 in [17].
a ν z ν is a polynomial of degree n having no zeros in |z| < k, k > 0, then: where, δ = ka 1 na 0 has absolute value ≤ 1 (see Lemma 1 in [9]). Lemma 11. Let p(z) = n ∑ ν=0 a ν z ν be a polynomial of degree n having no zeros in |z| < k, k > 0, then: and also: In particular, we have for r = 1 and R = k, where E t,k and λ are given by (21) and (12) respectively.

Main Results
In this paper, by involving certain coefficients of the polynomial, we obtain a generalization and refinement of Theorem 4. Moreover, our results are found to generalize and improve or generalize some other known inequalities. More precisely, we prove: a ν z ν be a polynomial of degree n having no zeros in |z| < k, k > 0, and where E t,k and F R,k are given by (21) and (42) respectively. Equality occurs in (45) when s = 0, k = 1 = r for p(z) = z n + 1 and in (46) when s = 1, r = R for p(z) = (z + k) n .
Taking r = 1 in Theorem 5, we have the following improvement of Theorem 4 proven by Jain [14].
a ν z ν be a polynomial of degree n having no zeros in |z| < k, k ≥ 1, and Remark 2. In the light of (40) of Lemma 11 and (41) of Lemma 12, Corollary 1 is an improvement of Theorem 4 due to Jain [14].  a ν z ν be a polynomial of degree n having no zeros in |z| < k, k > 0, and Inequality (49) reduces to inequality (5) when we set r = k = 1, and therefore is a generalization of (5) due to Ankeny and Rivlin [6]. Whereas, by inequality (38) of Lemma 11, inequality (50) is an improvement of a result proven by Jain Lemma 6 in [18]. Moreover, (50) is also an improvement of another result proven by Jain Remark 1 in [17].
Remark 5. If s = 0 and k = 1 in Theorem 5, we have the next result which generalizes Theorem 1 proven by Ankeny and Rivlin [6] and improves Theorem 3 due to Govil [8].
Again putting k = 1 in Corollary 2, we have: a ν z ν is a polynomial of degree n having no zeros in |z| < 1, then: Remark 6. If we put r = 1, inequality (51) reduces to (5) due to Ankeny and Rivlin [6]. For k = 1, inequality (38) gives: which shows that the bound of (52) improves over the bound given by inequality (7) due to Govil [8].

Remark 7.
If we take s = 0 and r = 1 in Theorem 5, we have the following result which improves the result proven by Aziz and Mohammad Theorem 2 in [19].
a ν z ν is a polynomial of degree n having no zeros in |z| < k, k ≥ 1, then: Inequality (53) is a generalization of (5). Taking r = 1 in (38), we obtain: which verifies that inequality (54) is an improvement of the result mentioned in Remark 7.

Remark 8.
Furthermore, if we take s = 0 and R = k = 1 in Theorem 5, the two inequalities reduce to a single inequality which is found to extend and improve inequality (6) due to Rivlin [7] in the form of the following result.
If s = 1, Theorem 5 in particular reduces to the following result which has some implications on some known inequalities involving the derivative of p(z).

Corollary 6.
If p(z) = n ∑ ν=0 a ν z ν is a polynomial of degree n having no zeros in |z| < k, k > 0, then: and Remark 9. If we choose r = 1 in Corollary 6, we have the next result which is an improvement of the result due to Bidkham and Dewan Theorem 3 in [10].
a ν z ν is a polynomial of degree n having no zeros in |z| < k, k ≥ 1, then: and M(p , R) ≤ n R + k|λ| Putting r = 1 in (38) of Lemma 11, we have: We can show that: which is equivalent to: and it simplifies to: which clearly holds as |λ| ≤ 1 and R ≤ k.
Thus, inequality (59) is an improvement of the result mentioned in Remark 9. On the other hand, inequality (58) gives the corresponding bound when R ≥ k.
We further improve Theorem 5 by involving m(p, k). In fact, we prove: a ν z ν be a polynomial of degree n ≥ 3 having no zeros in |z| < k, and where E t,k and F R,k are respectively given by (21) and (42). Equality occurs in (60) when s = 0, k = 1 = r for p(z) = z n + 1 and in (61) when s = 1, r = R for p(z) = (z + k) n .
The following result which further improves Corollary 1, is obtained by taking r = 1 in Theorem 6. Corollary 8. Let p(z) = n ∑ ν=0 a ν z ν be a polynomial of degree n ≥ 3 having no zeros in |z| < k, k ≥ 1, then for 0 ≤ s < n, and where E t,k and F R,k are respectively given by (21) and (42).
Corollary 8 has special importance that it gives Ankeny and Rivlin's analogue of the s th derivative separately by two different forms of bound according as R ≥ k and 1 ≤ R ≤ k.
Remark 11. In view of (39), inequality (62) gives a better bound than a result due to Mir Theorem 3 in [13] concerning the estimate of M(p (s) , R). Whereas, inequality (63) gives an improved and generalized version of the result of Bidkham and Dewan Theorem 3 in [10] concerning s th derivative.
Remark 13. Putting s = 0, R = k = 1 in Theorem 6, the two inequalities coincide to the following result which further extends and improves (6) due to Rivlin [7].
a ν z ν is a polynomial of degree n ≥ 3 having no zeros in |z| < 1, then: where E t,1 is obtained by taking k = 1 in (21).

Proofs of the Theorems
We first prove the more refined Theorem 6.
Proof of Theorem 5 . The proof of Theorem 5 follows along the lines of Theorem 6, on applying (20) of Lemma 7 instead of applying (31) of Lemma 8.

Conclusions
The well-known result due to Ankeny and Rivlin (Theorem 1), and some related results are generalized and extended by proving inequalities for the s th derivative involving certain coefficients of the polynomials. In addition, the result due to Rivlin (Theorem 2) and some related results are generalized, improved, and extended in a similar manner. It would be of interest to further extend these results to polar derivatives of a higher order and L p norm inequalities.
Funding: This research received no external funding.
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