The Number of Zeros of a Polynomial in a Disk as a Consequence of Restrictions on the Coefficients

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1 The Number of Zeros of a Polynomial in a Disk as a Consequence of Restrictions on the Coefficients Robert Gardner Brett Shields Department of Mathematics and Statistics Department of Mathematics and Statistics East Tennessee State University East Tennessee State University Johnson City, Tennessee Johnson City, Tennessee gardnerr@etsu.edu shieldsb@goldmail.etsu.edu Abstract. In this paper, we put restrictions on the coefficients of polynomials and give bounds concerning the number of zeros in a specific region. The restrictions involve a monotonicity-type condition on the coefficients of the even powers of the variable and on the coefficients of the odd powers of the variable (treated separately). We present results by imposing the restrictions on the moduli of the coefficients, the real and imaginary parts of the coefficients, and the real parts (only) of the coefficients. AMS Mathematics Subject Classification (200): 30A0 Keywords: polynomials, counting zeros, monotone coefficients. Introduction The classical Eneström-Kakeya Theorem restricts the location of the zeros of a polynomial based on a condition imposed on the coefficients of the polynomial. Namely: n Eneström-Kakeya Theorem. Let P(z) = a j z j be such that 0 < a 0 a a 2... a n a n. Then all the zeros of P lie in z. There exists a huge body of literature on these types of results. For a brief survey, see Section of Topics in Polynomials: Extremal Problems, Inequalities, Zeros by Milovanović,

2 Mitrinović, and Rassias [8]. A more detailed and contemporary survey is to appear in [5]. In 996, Aziz and Zargar [2] introduced the idea of imposing a monotonicity condition on the coefficients of the even powers of z and on the coefficients of the odd powers of z separately in order to get a restriction on the location of the zeros of a polynomial with positive real coefficients (see their Theorem 3). These types of hypotheses apply to more polynomials than a simple restriction of monotonicity on the coefficients as used in the Eneström-Kakeya Theorem. Of course if the hypotheses of the Eneström-Kakeya Theorem are satisfied by a polynomial P(z) = n a jz j, say 0 < a 0 a a 2... a n a n, then P also satisfies the hypotheses of Aziz and Zargar: 0 < a 0 a 2 a 4... a and 0 < a a 3... a 2 (n)/2. Notice that is the largest even subscript and 2 (n )/2 is the largest odd subscript, regardless of the parity of n. However, a polynomial such as p(z) = 2 z 3z 2 2z 3 4z 4 3z 5 5z 6 4z 7 does not satisfy the monotonicity condition of Eneström-Kakeya, but does satisfy the Aziz and Zargar hypotheses. Therefore Aziz and Zargar s hypotheses are less restrictive and are satisfied by a larger class of polynomials. Cao and Gardner [3] generalized this idea of imposing a monotonicity condition on even and odd indexed coefficients separately and imposed the following conditions on the coefficients of a polynomial P(z) = n a jz j where Re(a j ) = α j and Im(a j ) = β j : α 0 α 2 t 2 α 4 t 4 α 2k t 2k α 2k2 t 2k2 α t, α α 3 t 2 α 5 t 4 α 2l t 2l 2 α 2l t 2l α 2 (n)/2 t, β 0 β 2 t 2 β 4 t 4 β 2s t 2s β 2s2 t 2s2 β t, β β 3 t 2 β 5 t 4 β 2q t 2q 2 β 2q t 2q β 2 (n)/2 t, for some k, l, s, q between 0 and n. Again, Cao and Gardner presented results restricting the location of the zeros of P. In this paper, we impose these monotonicity-type conditions on the coefficients of a polynomial and then give a restriction on the number of zeros of the polynomial in a disk of a specific radius which is centered at zero. The first result relevant to this study which involves counting zeros appears in Titchmarsh s 939 The Theory of Functions []: 2

3 Theorem. Let F(z) be analytic in z R. Let F(z) M in the disk z R and suppose F(0) 0. Then for 0 < δ < the number of zeros of F in the disk z δr does not exceed log /δ log M F(0). Notice that in order to apply Theorem., we must find max z R F(z). Since this can be, in general, very complicated, it is desirable to present results which give bounds on the number of zeros in terms of something more tangible, such as the coefficients in the series expansion of F. With the same hypotheses as the Eneström-Kakeya Theorem, Mohammad used a special case of Theorem. to prove the following [9]. Theorem.2 Let P(z) = n a j z j be such that 0 < a 0 a a 2 a n a n. Then the number of zeros in z 2 does not exceed log 2 log ( an a 0 ). Theorem.2 was extended from polynomials with real coefficients to polynomials with complex coefficients by K. K. Dewan [4, 8]. Using hypotheses related to those of Theorem.2, Dewan imposed a monotonicity condition on the moduli and then on the real parts of the coefficients. Her results were recently generalized by Pukhta [0] who proved the following two theorems. n Theorem.3 Let P(z) = a j z j be such that arg(a j ) β α π/2 for 0 j n and for some real α and β, and let 0 < a a 2 a n a n. Then for 0 < δ < the number of zeros of P in z δ does not exceed log /δ log a n (cosα sinα ) 2sin α n a j. 3

4 Theorem.4 Let P(z) = n a j z j be such that arg(a j ) β α π/2 for 0 j n and for some real α and β, and let 0 < α 0 α α 2 α n α n. Then for 0 < δ < the number of zeros of P in z δ does not exceed log /δ log α n n k=0 β k. In a result similar to the Eneström-Kakeya Theorem, but for analytic functions as opposed to polynomials, Aziz and Mohammad [] imposed the condition 0 < α 0 tα t k α k t k α k on the real parts (and a similar condition on the imaginary parts) of the coefficients of analytic function F(z) = a jz j. Gardner and Shields [6] used this type of hypothesis to prove the following three results. Theorem.5 Let P(z) = n a j z j where for some t > 0 and some 0 k n, 0 < t a t 2 a 2 t k a k t k a k t k a k t n a n t n a n and arg(a j ) β α π/2 for j n and for some real α and β. Then for 0 < δ < the number of zeros of P in the disk z δt does not exceed where log /δ log M M = (t a n t n )( cos α sinα) 2 a k t k cosα 2sin α n a j t j. Theorem.6 Let P(z) = n a j z j where Re(a j ) = α j and Im(a j ) = β j for 0 j n. Suppose that for some t > 0 and some 0 k n we have 0 α 0 tα t 2 α 2 t k α k t k α k t k α k t n α n t n α n. Then for 0 < δ < the number of zeros of P in the disk z δt does not exceed log /δ log M, where M = ( α 0 α 0 )t 2α k t k ( α n α n )t n 2 n β j t j. 4

5 Theorem.7 Let P(z) = n a j z j where Re(a j ) = α j and Im(a j ) = β j for 0 j n. Suppose that for some t > 0, for some 0 k n we have 0 α 0 tα t 2 α 2 t k α k t k α k t k α k t n α n t n α n, and for some 0 l n we have β 0 tβ t 2 β 2 t l β l t l β l t l β l t n β n t n β n. Then for 0 < δ < the number of zeros of P in the disk z δt does not exceed log /δ log M, where M = ( α 0 α 0 )t 2α k t k ( α n α n )t n ( β 0 β 0 )t 2β l t l ( β n β n )t n. The purpose of this paper is to put the monotonicity-type condition of Cao and Gardner on () the moduli, (2) the real and imaginary parts, and (3) the real parts only of the coefficients of polynomials. In this way, we give results related to Theorems.5,.6, and.7, but with more flexible hypotheses and hence applicable to a larger class of polynomials. 2 Results We now present three major theorems, as described above, and several corollaries. First, we put the monotonicity-type condition on the moduli of the coefficients. Theorem 2. Let P(z) = and s n we have n a j z j. Suppose that for some t > 0, and for some 0 k n 0 a 2 t 2 a 4 t 4 a 2k t 2k a 2k2 t 2k2 a t, a a 3 t 2 a 5 t 4 a 2s t 2s 2 a 2s t 2s a 2 (n)/2 t Then for 0 < δ < the number of zeros of P in the disk z δt does not exceed log /δ log M 5

6 where M = (t 2 a t 3 a n t n a n t n2 )( cosα sinα) n 2cos α( a 2k t 2k2 a 2s t 2s ) 2sin α a j t j2. When t =, Theorem 2. yields the following. Corollary 2.2 Let P(z) = have n a j z j. Suppose that for some 0 k n and s n we 0 a 2 a 4 a 2k a 2k2 a, a a 3 a 5 a 2s a 2s a 2 (n)/2. Then for 0 < δ < the number of zeros of P in the disk z δ does not exceed where log /δ log M M = ( a a n a n )( cos α sinα) 2cos α( a 2k a 2s ) 2sin α n a j. Next, we put the monotonicity-type condition on the real and imaginary parts of the coefficients. n Theorem 2.3 Let P(z) = a j z j where Re(a j ) = α j, Im(a j ) = β j for 0 j n. Suppose that for some t > 0, some 0 k n, l n, 0 s n, and q n we have 0 α 0 α 2 t 2 α 4 t 4 α 2k t 2k α 2k2 t 2k2 α t, α α 3 t 2 α 5 t 4 α 2l t 2l 2 α 2l t 2l α 2 (n)/2 t, β 0 β 2 t 2 β 4 t 4 β 2s t 2s β 2s2 t 2s2 β t, β β 3 t 2 β 5 t 4 β 2q t 2q 2 β 2q t 2q β 2 (n)/2 t. 6

7 Then for 0 < δ < the number of zeros of P in the disk z δt does not exceed where log /δ log M M = ( α 0 α 0 β 0 β 0 )t 2 ( α α β β )t 3 2(α 2k t 2k2 α 2l t 2l β 2s t 2s2 β 2q t 2q ) ( α n α n β n β n )t n ( α n α n β n β n )t n2. With t = in Theorem 2.3, we have: n Corollary 2.4 Let P(z) = a j z j where Re(a j ) = α j, Im(a j ) = β j for 0 j n. Suppose that for some 0 k n, l n, 0 s n, and q n we have 0 α 0 α 2 α 4 α 2k α 2k2 α, α α 3 α 5 α 2l α 2l α 2 (n)/2, β 0 β 2 β 4 β 2s β 2s2 β, β β 3 β 5 β 2q β 2q β 2 (n)/2. Then for 0 < δ < the number of zeros of P in the disk z δ does not exceed where log /δ log M M = ( α 0 α 0 β 0 β 0 ) ( α α β β ) 2(α 2k α 2l β 2s β 2q ) ( α n α n β n β n ) ( α n α n β n β n ). By manipulating the parameters k, l, s, and q we easily get eight corollaries from Corollary 2.4. For example, with k = s = and l = q = 2 (n )/2 we have: 7

8 Corollary 2.5 Let P(z) = that: n a j z j where Re(a j ) = α j, Im(a j ) = β j for 0 j n. Suppose 0 α 0 α 2 α 4 α, α α 3 α 5 α 2 (n)/2 β 0 β 2 β 4 β, β β 3 β 5 β 2 (n)/2. Then for 0 < δ < the number of zeros of P in the disk z δ does not exceed where log /δ log M M = ( α 0 α 0 β 0 β 0 )( α α β β )( α n α n β n β n )( α n α n β n β n ). With k =, l =, and each a j real in Corollary 2.4 we have: Corollary 2.6 Let P(z) = n a j z j where a j R for 0 j n. Suppose that: 0 a 0 a 2 a 4 a, a a 3 a 5 a 2 (n)/2. Then for 0 < δ < the number of zeros of P in the disk z δ does not exceed log /δ log M where M = a 0 a 0 a a. Example. Consider the polynomial P(z) = 0z 2z 2 0z 3 3z 4 0z 5 4z 6 0z 7 8z 8. The zeros of P are approximately 0.029, , ± i, ±.05362i, and ± i. Applying Corollary 2.6 with δ = 0.5 we see that it predicts no more than zeros in z 0.5. In other words, Corollary 2.6 predicts at most one zero in z 0.5. In fact, P does have exactly one zero in z 0.5, and Corollary 2.6 is sharp for this example. 8

9 n Theorem 2.7 Let P(z) = a j z j where Re(a j ) = α j, Im(a j ) = β j for 0 j n. Suppose that for some t > 0, some 0 k n and l n we have 0 α 0 α 2 t 2 α 4 t 4 α 2k t 2k α 2k2 t 2k2 α t, α α 3 t 2 α 5 t 4 α 2l t 2l 2 α 2l t 2l α 2 (n)/2 t Then for 0 < δ < the number of zeros of P in the disk z δt does not exceed log /δ log M where M = ( α 0 α 0 )t 2 ( α α )t 3 2α 2k t 2k2 α 2l t 2l ( α n α n )t n ( α n α n )t n2 2 n β j t j2. With t = in Theorem 2.7, we get the following: n Corollary 2.8 Let P(z) = a j z j where Re(a j ) = α j, Im(a j ) = β j for 0 j n. Suppose that for some 0 k n and l n we have 0 α 0 α 2 α 4 α 2k α 2k2 α, α α 3 α 5 α 2l α 2l α 2 (n)/2 Then for 0 < δ < the number of zeros of P in the disk z δ does not exceed where log /δ log M M = ( α 0 α 0 ) ( α α ) 2α 2k α 2l ( α n α n ) ( α n α n ) 2 n β j. 9

10 3 Proofs of the Results The following is due to Govil and Rahman and appears in [7]. Lemma 3. Let z, z C with z z. Suppose argz β α π/2 for z {z, z } and for some real α and β. Then z z ( z z )cos α ( z z )sinα. We now give proofs of our results. Proof of Theorem 2.. Define G(z) = (t 2 z 2 )P(z) = t 2 a 0 a t 2 z For z = t we have G(z) t 2 a t 3 = t 2 a t 3 2s t 2 a t 3 k2 2s 2 (n)/2 n (a j t 2 a j 2 )z j a n z n a n z n2. n a j t 2 a j 2 t j a n t n a n t n2 2k a j t 2 a j 2 t j 2k α j t 2 α j 2 t j 2 (n)/2 s k2 a j t 2 a j 2 t j a j t 2 a j 2 t j a n t n a n t n2 {( a j t 2 a j 2 )cosα ( a j 2 a j t 2 )sin α}t j {( a j 2 a j t 2 )cos α ( a j 2 a j t 2 )sinα}t j {( a j t 2 a j 2 )cos α ( a j 2 a j t 2 )sinα}t j s {( a j 2 a j t 2 )cos α ( a j 2 a j t 2 )sinα}t j 0

11 a n t n a n t n2 by Lemma 3. = t 2 a t 3 t 2 cos α a 2k t 2k2 cosα t 2 sinα a 2k t 2k2 sinα 2sin α 2k 2 a j t j2 a 2k t 2k2 cos α a t 2 cosα 2 a 2k t 2k2 sinα a t 2 sinα 2sin α a j t j2 k2 a t 3 cos α a 2s t 2s cosα a t 3 sinα a 2s t 2s sinα 2sin α a 2s t 2s cos α a 2 (n)/2 t 2 (n)/2 cos α a 2s t 2s sinα 2s 3 2 (n)/2 3 a 2 (n)/2 t 2 (n)/2 sinα 2sinα a j t j2 a n t n a n t n2 s = t 2 ( cosα sinα) a t 3 ( cosα sinα) 2 a 2k t 2k2 (cosα sinα) 2 a 2s t 2s (cosα sin α) 2sin α 2sin α 2s 2 2k 2 2 (n)/2 3 a j t j2 2sin α a j t j2 s a n t n ( cos α sinα) a n t n2 ( cos α sinα) = (t 2 a t 3 a n t n a n t n2 )( cos α sinα) n 2cos α( a 2k t 2k2 a 2s t 2s ) 2sin α a j t j2 = M. 2 a j t j2 2sin α a j t j2 k2 Now G(z) is analytic in z t, and G(z) M for z = t. So by Titchmarsh s theorem (Theorem.) and the Maximum Modulus Theorem, the number of zeros of G (and hence of P) in z δt does not exceed The theorem follows. log /δ log M. a j t j2

12 Proof of Theorem 2.3. Define G(z) = (t 2 z 2 )P(z). For z = t we have G(z) ( α 0 β 0 )t 2 ( α 2 β )t 3 n α j t 2 α j 2 t j ( α n β n )t n ( α n β n )t n2 = ( α 0 β 0 )t 2 ( α 2 β )t 3 β j t 2 β j 2 t j 2 (n)/2 ( α n β n )t n ( α n β n )t n2 = ( α 0 β 0 )t 2 ( α 2 β )t 3 2l s2 (α j t 2 α j 2 )t j (β j 2 β j t 2 )t j α j t 2 α j 2 t j β j t 2 β j 2 t j 2k 2 (n)/2 l 2q n β j t 2 β j 2 t j (α j t 2 α j 2 )t j (α j 2 α j t 2 )t j (β j t 2 β j 2 )t j ( α n β n )t n ( α n β n )t n2 = ( α 0 β 0 )t 2 ( α 2 β )t 3 2k 2l 2s 2q α j t j2 α j t j2 β j t j2 β j t j2 2k 2l 2s 2q α j 2 t j α j 2 t j β j 2 t j β j 2 t j k2 2 (n)/2 l s2 ( α n β n )t n ( α n β n )t n2 2 (n)/2 q 2 (n)/2 k2 2s 2 (n)/2 q α j 2 t j β j 2 t j k2 α j 2 t j s2 β j 2 t j α j t 2 α j 2 t j (α j 2 α j t 2 )t j (β j t 2 β j 2 )t j (β j 2 β j t 2 )t j α j t j2 2 (n)/2 l β j t j2 2 (n)/2 q α j t j2 β j t j2 2

13 = ( α 0 α 0 β 0 β 0 )t 2 ( α α β β )t 3 = M. 2(α 2k t 2k2 α 2l t 2l β 2s t 2s2 β 2q t 2q ) ( α n α n β n β n )t n ( α n α n β n β n )t n2 The result now follows as in the proof of Theorem 2.. Proof of Theorem 2.7. Define G(z) = (t 2 z 2 )P(z). For z = t we have G(z) ( α 0 β 0 )t 2 ( α β )t 3 n α j t 2 α j 2 t j ( α n β n )t n ( α n β n )t n2 = ( α 0 β 0 )t 2 ( α β )t 3 2 (n)/2 α j t 2 α j 2 t j α j t 2 α j 2 t j n ( β j t 2 β j 2 )t j ( α n β n )t n ( α n β n )t n2 = ( α 0 β 0 )t 2 ( α β )t 3 k2 (α j 2 α j t 2 )t j n β j t 2 2k 2l 2k (α j t 2 α j 2 )t j (α j t 2 α j 2 )t j n ( β j t 2 β j 2 )t j 2 (n)/2 l (α j 2 α j t 2 )t j n β j 2 t j ( α n β n )t n ( α n β n )t n2 = ( α 0 β 0 )t 2 ( α β )t 3 2l α j t j2 α j t j2 2k 2l n n 2 β j t 2 β j t j2 α j 2 t j α j 2 t j k2 2 (n)/2 l α j 2 t j k2 α j 2 t j = ( α 0 α 0 )t 2 ( α α )t 3 2α 2k t 2k2 α 2l t 2l 3 α j t j2 2 (n)/2 l α j t j2

14 n ( α n α n )t n ( α n α n )t n2 2 β j t j2 = M. The result now follows as in the proof of Theorem 2.. References [] A. Aziz and Q. G. Mohammad, On the Zeros of a Certain Class of Polynomials and Related Entire Functions, Journal of Mathematical Analysis and Applications, 75 (980), [2] A. Aziz and B. A. Zargar, Some Extensions of Eneström-Kakeya, Glasnik Matematički, 3(5), (996). [3] J. Cao and R. Gardner, Restrictions on the Zeros of a Polynomial as a Consequence of Conditions on the Coefficients of even Powers and Odd Powers of the Variable, Journal of Computational and Applied Mathematics, 55, (2003). [4] K. K. Dewan, Extremal Properties and Coefficient Estimates for Polynomials with Restricted Zeros and on Location of Zeros of Polynomials, Ph.D. Thesis, Indian Institutes of Technology, Delhi, 980. [5] R. Gardner and N. K. Govil, The Eneström-Kakeya Theorem and Some of Its Generalizations, Current Topics in Pure and Computational Complex Analysis, Springer- Verlag, to appear. [6] R. Gardner and B. Shields, The Number of Zeros of a Polynomial in a Disk, Journal of Classical Analysis, 3(2) (203), [7] N. K. Govil and Q. I. Rahman, On the Eneström-Kakeya Theorem, Tôhoku Mathematics Journal, 20 (968), [8] G. V. Milovanović, D. S. Mitrinović, and Th. M. Rassias, Topics in Polynomials: Extremal Problems, Inequalities, Zeros, Singapore: World Scientific Publishing,

15 [9] Q. G. Mohammad, On the Zeros of the Polynomials, American Mathematical Monthly, 72(6) (965), [0] M. S. Pukhta, On the Zeros of a Polynomial, Applied Mathematics, 2 (20), [] E. C. Titchmarsh, The Theory of Functions, 2nd Edition, Oxford University Press, London,