Some remarks on rotation theorems for complex polynomials
aa r X i v : . [ m a t h . C V ] F e b Some remarks on rotation theoremsfor complex polynomials
V.N. Dubinin ∗ Far Eastern Federal University (FEFU), 8, Sukhanova Street, Vladivostok, 690950, RussiaInstitute of Applied Mathematics, FEBRAS, 7, Radio Street, Vladivostok, 690041, Russia
Abstract
For any complex polynomial P ( z ) = c + c z + ... + c n z n , c n = 0 , having all its zerosin the unit disk | z | ≤ , we consider the behavior of the function (arg P ( e iθ )) ′ θ whenthe real argument θ changes. We give some sharp estimates of this function involvingof the values of P ( e iθ ) , arg P ( e iθ ) or the coefficients c k , k = 0 , , n − , n. Keywords: complex polynomials, rotation theorems, inequalities, boundary Schwarzlemma, rational functions.MSC2010: 30A10; 30C10; 30C15; 30C80.
1. Introduction
Let all zeros of the polynomial P ( z ) = c + c z + ... + c n z n , c n = 0 , (1)lie in the unit disk | z | ≤ , c k ∈ C , k = 0 , , ..., n, n ≥ . In the theory of polynomials,inequality Re zP ′ ( z ) P ( z ) ≥ n , (2)is well known, which is valid for all points z on the circle | z | = 1 that are different from thezeros of the polynomial P . The equality in (2) takes place if and only if all zeros of P lieon | z | = 1 (see, for example, [1, p. 439]). Inequality (2) can be interpreted as a rotationtheorem for the complex polynomial P on the circle | z | = 1:(arg P ( z )) ′ θ ≥ n , (3) ∗ E-mail address: [email protected] = e iθ , ≤ θ ≤ π. In a series of papers beginning with [2], this author developed thegeometric function theory approach to inequalities for polynomials and rational functions.This is described in detail in the survey article [6] and some subsequent publications of theauthor. In particular, as an example of the application of the boundary Schwarz lemma, thefollowing strengthening of inequality (3) is given in [5](arg P ( z )) ′ θ ≥ n | c n | − | c | | c n | + | c | ) ≥ n , (4) | z | = 1 , P ( z ) = 0 . Equality in (4) holds for polynomials P with zeros lying on the unit circle | z | = 1 and for any z, | z | = 1 , P ( z ) = 0 [5, inequality (10)]. Note that if the polynomial P ( z )of the form (1) has no zeros in the disk | z | <
1, then the zeros of the polynomial z n P (1 /z )lie in the disk | z | ≤
1. Applying (4) to the last polynomial, we obtain the inequality(arg P ( z )) ′ θ ≤ n | c n | − | c | | c n | + | c | ) ≤ n , for all z on | z | = 1 for which P ( z ) = 0. Earlier [2], a weakened version of inequality (4) wasestablished, in which the right-hand side of (4) is replaced by the quantity n p | c n | − p | c | p | c n | ≥ n , [2, Theorem 4]. At present, the application of the boundary Schwarz lemma to inequalitiesfor complex polynomials had raised considerable interest (see, for example, [11], [12], [14],[18] – [23], [25] and references therein ). As regards inequality (4) directly, we note thefollowing results. In the papers of Govil, Kumar [10] and Rather, Dar, Iqbal [19] inequality(4) is generalized in different directions and another proofs of the above inequality are given.Gulzar, Zargar, Akhter [11], Hussain, Ahmad [12], and Milovanovi´c, Mir, and Ahmad [18]used (4) to generalize and strengthen the well-known inequalities for polynomials. In thepaper of Rather, Dar, Iqbal [20] the weakened version (4) was used.In this note, we, first, refine inequality (4) taking into account the values of the polynomial P on the circle | z | = 1. Further, the strengthening of (4) with the involvement of thecoefficients c and c n − of the polynomial P is obtained. In addition, we give an upperbound for the left-hand side of (3) in the form of a finite increment theorem. The proofs ofall theorems are carried out in a unified manner and are based on the geometric functiontheory approach [6]. In the final Section 3, some remarks are made concerning the estimationof the value (arg R ( z )) ′ θ for the rational function R .
2. Rotation theorems
For points z = e iθ different from the zeros of a polynomial P , we introduce the notationΛ( P, z ) = 2(arg P ( e iθ )) ′ θ − n. In view of (3) Λ(
P, z ) ≥
02n the circle | z | = 1. It is natural to pose the question of estimating the rotation speed P ( z )when z rotates on the circle | z | = 1, depending on the value of P ( z ) itself. Some progress inthe solution of this question is given by Theorem 1.
If all zeros of a polynomial P of the form (1) lie in the disc | z | ≤ , then forany point z a on the circle | z | = 1 that is different from zero of the polynomial P , inequality Λ( P, z ) ≥ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (Λ( P, z ) + 1) c P ( z ) c n z n P ( z ) − (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (5) holds. Equality in (5) at the point z = 1 is attained for a polynomial P of the form P ( z ) = ( z − α ) n Y k =2 ( z − α k ) , (6) where α k , k = 1 , ..., n, are arbitrary numbers that satisfy the relations | α | < , | α k | =1 , α k = 1 , k = 2 , ..., n. Proof.
Let P ( z ) = c n n Y k =1 ( z − α k ) , and let α k = 1 , k = 1 , ..., n. In [8] (see also [9, Theorem 1]), among other results, Goryainovobtained inequality (cid:12)(cid:12)(cid:12)(cid:12) f ′ (0) − f ′ (1) (cid:12)(cid:12)(cid:12)(cid:12) ≤ − f ′ (1) , (7)which is valid for any holomorphic map of the disk | z | < f (0) = 0 , f (1) = 1 and f ′ (1) = ∞ . Here f (1) is the angular limit of f when z →
1, and f ′ (1) is the angular derivative of the function f at the point z = 1. It is well known that f ′ (1) ≥ . Equality in (7) is attained for the function f ∗ ( z ) = z − a − a z − a − az for any a, | a | < . Consider the Blaschke product f ( z ) = c n c n " n Y k =1 − α k − α k P ( z ) z n − P (1 /z ) = z " n Y k =1 − α k − α k n Y k =1 z − α k − α k z . If | α | = 1, then z − α − αz = 1 α · z − α /α − z = − α , and for | α | < w = z − α − αz
3s a linear-fractional mapping of the disk | z | < | w | < , such that the circle | z | = 1 turns to the circle | w | = 1. Thus, the function f is holomorphic in the disk | z | < | f ( z ) | < | z | < , f (0) = 0 , f (1) = 1 and f ′ (1) ≥ . Taking into account the geometric meaning of the derivative, we conclude that at the pointsof the circle | z | = 1, different from the zeros of the polynomial P , | f ′ ( z ) | = zf ′ ( z ) f ( z ) = z n P (1 /z ) P ( z ) · z n − P ′ ( z ) P (1 /z ) − P ( z )[( n − z n − P (1 /z ) − z n − P ′ (1 /z )]( z n − P (1 /z )) == zP ′ ( z ) P ( z ) − n + 1 + P ′ ( z ) zP ( z ) = 2Re zP ′ ( z ) P ( z ) − n + 1is satisfied.Goryainov’s inequality (7) applied to the function f gives (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) c P (1) c n P (1) (cid:20) P ′ (1) P (1) − n + 1 (cid:21) − (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ P ′ (1) P (1) − n. In general, if point z , | z | = 1 , is different from the zeros of the polynomial P , then thepoint z = 1 is different from the zeros of the polynomial ˜ P ( z ) := P ( zz ). Applying whatwas proved above to the polynomial ˜ P , we have (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) c P ( z ) c n z n P ( z ) (cid:20) z P ′ ( z ) P ( z ) − n + 1 (cid:21) − (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ z P ′ ( z ) P ( z ) − n. The resulting inequality coincides with inequality (5) for z = z . If now P ( z ) has the form(6), then f ( z ) = f ∗ ( z ) , where a = α . Therefore, equality is attained in (7). Hence the equality holds in (5). This completes theproof of Theorem 1.Inequality (5) is stronger than (4). Indeed,Λ(
P, z ) ≥ − (Λ( P, z ) + 1) (cid:12)(cid:12)(cid:12)(cid:12) c c n (cid:12)(cid:12)(cid:12)(cid:12) is fulfilled from (5). Therefore, Λ( P, z ) ≥ | c n | − | c || c n | + | c | , which is equivalent to inequality (4). Theorem 2.
Let all zeros of a polynomial P of the form (1) lie in the disc | z | ≤ . Thenfor any point z on the circle | z | = 1 , different from the zeros of the polynomial P , inequality Λ( P, z ) ≥ | c | − | c n | ) | c n | − | c | + | c n c − c c n − | (8)4 olds . Equality in (8) is attained for polynomials P with zeros on the unit circle | z | = 1 forany z, | z | = 1 , P ( z ) = 0 . Proof.
The equality | c | = | c n | holds if and only if all zeros of the polynomial P lie onthe circle | z | = 1. Therefore, the case of equality in (8) is obvious. Further, we consider that | c | 6 = | c n | . We need the following inequality (7) from [4]: | f ′ ( z ) | ≥ − | f ′ (0) | ) − | f ′ (0) | + | f ′′ (0) / | . (9)Here f is a holomorphic map of the disk | z | < f (0) = 0 , | f ′ (0) | 6 = 1 and z is an arbitrary point on the circle | z | = 1, in which the derivative f ′ ( z ) exists and | f ( z ) | = 1. In [16], Mercer presented a direct proof of (9) using Rogosinski’slemma (see also [17, inequality (15)]). Consider the Blaschke product f ( z ) = P ( z ) z n − P (1 /z ) = c n zc n n Y k =1 z − α k − α k z , where P ( z ) = c n Q nk =1 ( z − α k ) . As above, we see that f is a holomorphic map of the disk | z | < f (0) = 0 , and at each point z on the circle | z | = 1, nonzero of thepolynomial P , | f ( z ) | = 1 , and there is a derivative f ′ ( z ). Moreover, f ′ (0) = c n c n n Y k =1 ( − α k ) = c c n , so | f ′ (0) | 6 = 1 . Note that the derivative is n Y k =1 z − α k − α k z ! ′ = n Y k =1 z − α k − α k z ! n X k =1 (cid:18) z − α k + α k − α k z (cid:19) . In view of this and Vieta’s formulas, we have f ′′ (0) = 2 c n c n n Y k =1 ( − α k ) n X k =1 (cid:18) − α k + α k (cid:19) = 2 c c n (cid:20) c c − c n − c n (cid:21) . Substituting the found values of the derivatives into (9), taking into account the calculationsof the | f ′ ( z ) | in the proof of Theorem 1, we arrive at inequality (8). The theorem is proved.As noted by Merser [17, Remark 3.2] for the function f from (9) | f ′′ (0) | ≤ − | f ′ (0) | )is fulfilled. This attracts | c c n − c c n − | ≤ | c n | − | c | . Hence it is easy to see that inequality (8) is stronger (4). In the case of | c | = | c n | , the expression on the right-hand side of (8) means zero. heorem 3. Let all zeros of a polynomial P of degree n lie in the disc | z | ≤ . Supposethat for some point z on the circle | z | = 1 , the arc of this circle γ ( z , α ) := { z : | z | = 1 , | arg z − arg z | < α } , < α < π, does not contain the zeros of the polynomial P and the increment of the value P ( z ) − n arg z along any curve on the arc γ ( z , α ) with an end point at z does not exceed in absolute value β, < β < π . Then Λ( P, z ) ≤ tg β tg α . (10) Equality in (10) is attained for a polynomial P of the form P ( z ) = c n z ( z − α ) ... ( z − α n ) , (11) | α k | = 1 , α k = 1 , k = 2 , ..., n, point z = 1 , any α, < α < π, for which the γ (1 , α ) doesnot contain α k , k = 2 , ..., n and β = α . Proof.
We can assume that the point z = 1. In [7], among other results, inequality | f ′ (1) | ≤ tg β tg α . (12)was established. Here f is a holomorphic self-mapping of the unit disk | z | < f ( γ (1 , α )) ⊂ γ (1 , β ) for some α and β , 0 < α < π, < β < π, and f has an angular limit f (1) = 1 anda finite angular derivative f ′ (1). Equality in (12) is attained for the function f ( z ) ≡ z and α = β . We put as above P ( z ) = c n n Y k =1 ( z − α k ) . Let us show that the Blaschke product f ( z ) := c n c n " n Y k =1 − α k − α k P ( z ) z n P (1 /z ) = " n Y k =1 − α k − α k n Y k =1 z − α k − α k z . satisfies the conditions for inequality (12). Indeed, f is a holomorphic function mappingthe disk | z | < f is differentiable on the arc γ (1 , α ) , f (1) = 1 and | f ( z ) | = 1 on γ (1 , α ). In view of the hypothesis of Theorem 3, the increment of the argumentof the function f along any curve γ on γ (1 , α ) with an end point at the point z = 1 does notexceed in absolute value β : | ∆ γ arg f ( z ) | = | ∆ γ (2arg P ( z ) − n arg z ) | ≤ β. Hence, f ( γ (1 , α )) ⊂ γ (1 , β ) . Following the calculations carried out in the proof of Theorem1, we see that | f ′ (1) | = 2Re P ′ (1) P (1) − n. P ′ (1) P (1) − n ≤ tg β tg α , which is equivalent to (10) for z = 1. If now P ( z ) has the form (11), then f ( z ) ≡ z. In thiscase, f ′ (1) = 1 and | ∆ γ (2arg P ( z ) − n arg z ) | = | ∆ γ arg z | ≤ α, γ ⊂ γ (1 , α ) . and you can take β = α . This gives equality in (12) and hence equality in (10). The theoremis proved.
3. On rational functions
In conclusion we consider an application of the function theory to inequalities for rationalfunctions with prescribed poles: R ( z ) = P ( z ) Q nk =1 ( z − a k ) , where P ( z ) is an algebraic polynomial of degree m and | a k | > , k = 1 , ..., n. In someproblems, extremal functions are related to the Blaschke product B ( z ) = n Y k =1 − a k zz − a k , which is in general defined for any system of poles ( a , ..., a n ) , | a k | 6 = 1 , k = 1 , ..., n. The following analogue of the polynomial inequality (3) is known. If a rational function R has exactly m zeros (counting multiplicities) belonging to the disk | z | ≤
1, then(arg R ( z )) ′ θ ≥
12 ( m − n + (arg B ( z )) ′ θ ) (13)for all points on the circle | z | = 1 different from zeros of R . If all zeros of R lie in thecomplement to the disk | z | <
1, then the reverse inequality is valid:(arg R ( z )) ′ θ ≤
12 ( m − n + (arg B ( z )) ′ θ ) . (14)Equality in (13), (14) holds for R ( z ) = αB ( z ) + β with | α | = | β | . The proof of inequalities(13) and (14) can be found in [15, Lemma 4], [3, Lemma 3 and Theorem 4], and in [3]the geometric function theory was first applied to such a range of problems. Using variousversions of the boundary Schwarz lemma, Wali and Shah [24], [26] strengthened (13), (14) indifferent directions. Kalmykov [13] recently proved two- and three-point distortion theoremsfor rational functions that generalize some known results on Bernstein-type inequalities forpolynomials and rational functions. The rational functions under study have either majo-rants or restrictions on location of their zeros. The proofs in [13] are based on the newversion of the Schwarz Lemma and univalence condition for holomorphic functions.7n this regard, it can be assumed that the application of the methods of geometricfunction theory will lead to new rotation theorems for rational functions with prescribedpoles, in particular, to theorems generalizing Theorems 1–3 of this article. Funding:
This work was supported by the Russian Basic Research Fund [grant number20-01-00018].
References [1] P. Borwein, T. Erd ′′