Some extensions and generalizations of Eneström Kakeya theorem

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Available online at wwwsciencedirectcom ScienceDirect Journal of the Nigerian Mathematical Society xx (xxxx) xxx xxx wwwelseviercom/locate/jnnms Some extensions and generalizations of Eneström Kakeya

1 Available online at wwwsciencedirectcom ScienceDirect Journal of the Nigerian Mathematical Society xx (xxxx) xxx xxx wwwelseviercom/locate/jnnms Some extensions and generalizations of Eneström Kakeya theorem Q AA Mogbademu a,, S Hans b, JA Adepoju c a Department of Mathematics, University of Lagos, Akoka Lagos, Nigeria b Department of Applied Science, ITM University, Gurgaon, India c Federal University of Petroleum, Effurum, Delta, Nigeria Received July 04; received in revised form March 05; accepted March 05 Dedicated to dearest mother Mrs FM Mogbademu on her 0th Birthday Abstract In this paper, we put restrictions on the coefficients of a polynomial in order to improve the bounds for their zeros in a specific 4 region Our results extend and generalise a number of previously well known theorems including Eneström Kakeya theorem 5 c 05 Production and Hosting by Elsevier BV on behalf of Nigerian Mathematical Society This is an open access article under 6 the CC BY-NC-ND license ( Keywords: Polynomials; Zeros; Eneström Kakeya Introduction 9 Let P(z) = n j=0 a j z j be a polynomial of degree n One of the fundamental problem of finding out the region Q 0 which contains all or a prescribed number of zeros of a polynomial was first studied by Gauss [] He proved: Theorem If P(z) = z n + n a j z j, where a j are all real, then P(z) has all its zeros in z R, where (i) R = max(, s), s being the sum of positive a j (ii) R = max(n a j ) j In 9, Cauchy [] gave more exact bounds for the moduli of zeros of a polynomial than those given by Gauss [] 4 He proved the following result 5 Theorem All the zeros of the polynomial P(z) = n j=0 a j z j of degree n lie in the circle z R, where R is the 6 root of the equation a 0 + a z + a z + + a n z n + + a n z n = 0 9 Peer review under responsibility of Nigerian Mathematical Society Corresponding author addresses: amogbademu@unilagedung (AA Mogbademu), sunilhans@gmailcom (S Hans), jadi0@yahoocom (JA Adepoju) / c 05 Production and Hosting by Elsevier BV on behalf of Nigerian Mathematical Society This is an open access article under the CC BY-NC-ND license (

2 AA Mogbademu et al / Journal of the Nigerian Mathematical Society xx (xxxx) xxx xxx Several generalisations and improvements of this result are available in the literature (see [,4]) The following elegant results on the location of zeros of a polynomial with restricted coefficients is known as the Eneström Kakeya theorem [5,6] Theorem (Eneström Kakeya) Let P(z) = n j=0 a j z j be a polynomial of degree n whose coefficients a j satisfy a n a n a a 0 > 0, the closed unit disk z Joyal, Labella and Rahman [4] extended Theorem to polynomials whose coefficients are monotonic but need not be non-negative as follows: Theorem 4 Let P(z) = n j=0 a j z j be a polynomial of degree n such that a n a n a a 0, z a n + a 0 a 0 a n Aziz and Zargar [] relaxed the conditions of Theorem and proved the following generalisation of Theorem 4 Theorem 5 Let P(z) = n j=0 a j z j be a polynomial of degree n such that for some k, ka n a n a a 0, z + k ka n + a 0 a 0 a n Govil and Rahman [] considered polynomials whose coefficients are not necessarily real Infact, they proved the following generalisation of Theorem Theorem 6 Let P(z) = n j=0 a j z j be a polynomial of degree n with Re(a j ) = α j and I m(a j ) = β j, j = 0,,,,n such that α n α n α α 0 > 0, where α n > 0, then P(z) has all its zeros in z + n β j α j=0 The following generalizations of Theorems 4, 5 and 6 was proved by Govil and Mc-tume [] Theorem Let P(z) = n j=0 a j z j be a polynomial of degree n with Re(a j ) = α j and I m(a j ) = β j, j = 0,,,,n such that for some k, kα n α n α α 0, where α n > 0, then P(z) has all its zeros in z + k kα n α 0 + α 0 + n β j α n j=0 Aziz and Zargar [9] obtained some extensions of Theorem by relaxing the hypothesis as follows:

3 AA Mogbademu et al / Journal of the Nigerian Mathematical Society xx (xxxx) xxx xxx Theorem Let P(z) = n j=0 a j z j be a polynomial of degree n If for some positive numbers k and ρ with k and 0 < ρ, ka n a n a ρa 0 0, the closed unit disk 4 z + k k + a 0 a n ( ρ) 5 Theorem 9 Let P(z) = n j=0 a j z j be a polynomial of degree n If for some positive number ρ, 0 < ρ, and 6 some non-negative integer λ, 0 λ n, a n a n a λ+ a λ a λ ρa 0, 9 z + a n a a λ a n + ( ρ) a 0 ρa 0 0 n a n In this paper, we further weaken the hypothesis of Theorems and 9 by considering a general class of polynomials to prove some extensions and generalisations of Theorem (Eneström Kakeya), which in turn improve the bounds in some cases Main results 4 Theorem Let P(z) = n j=0 a j z j be a polynomial of degree n with complex coefficients If Re(a j ) = α j and 5 I m(a j ) = β j, for 0 j n such that for some real t > 0, µ 0, λ, 0 λ n and ρ, 0 < ρ, 6 t n α n + µt n t n α n t λ+ α λ+ t λ α λ t λ α λ α 0, z µ α λ a n a n t n λ + µ α nt + ( α 0 a 0 ) + β 0 n t n + t n j 9 Proof Proof follows from our next Theorem as a special case 0 Notice that with t = in Theorem and, suppose the imaginary parts of the coefficients are monotonic and non- negative, then we get the following Corollary Let P(z) = n j=0 a j z j be a polynomial of degree n with complex coefficients If Re(a j ) = α j and I m(a j ) = β j, for 0 j n such that for some real k, 0 λ n and 0 < ρ, 4 α n + µ α n α λ+ α λ α λ α 0, 5 and 6 β n β n β β 0 > 0, z µ a n a n (α λ + µ α n + ( α 0 α 0 ) + β n ) 9 Take λ = n, µ = (k )α n, α 0 > 0 and β j = 0, for 0 j n in Corollary, the hypothesis becomes 0 kα n α n α 0 > 0,

4 4 AA Mogbademu et al / Journal of the Nigerian Mathematical Society xx (xxxx) xxx xxx therefore we obtain the main result of Aziz and Zargar [9] which inturn is an extension of Theorem (Eneström Kakeya) We state and prove the following result which is a generalisation of Theorem which itself is a generalisation of several results in the literature [0,], [,,, 5] and so on Theorem Let P(z) = n j=0 a j z j be a polynomial of degree n with complex coefficients If Re(a j ) = α j and I m(a j ) = β j, for 0 j n such that for some real t > 0, µ 0, λ, 0 λ n and ρ, 0 < ρ, t n α n + µt n t n α n t λ+ α λ+ t λ α λ t λ α λ ρα 0, z µ α λ a n a n t n λ + µ α nt + α 0 ρ( α 0 + α 0 ) + β 0 n t n + Proof Consider the polynomial F(z) = (t z)p(z) n = a 0 t + (a j t a j )z j a n z n+ = a n z n+ + n (α j t α j )z j + α 0 t + iβ 0 t + n (β j t β j )z j = a n z n+ + (α n t α n )z n n + (α j t α j )z j + α 0 t + iβ 0 t + = a n z n+ + (µ α n t)z n + α n tz n + (α n t µ α n )z n n + (α j t α j )z j + α 0 t + iβ 0 t + i n (β j t β j )z j This gives F(z) = a nz n+ + (µ α n t)z n + α n tz n + (α n t µ α n )z n Now, let z t, so that t n j n (β j t β j )z j + (α n t α n )z n + + (α λ+ t α λ )z λ+ + (α λ t α λ )z λ + + (α t α 0 )z n + (ρα 0 α 0 )z + α 0 t + iβ 0 t + i (β j t β j )z j z n a n z µ z n α n t µ α n + α n t α n z + + α λ+t α λ z n λ + α λt α λ z n λ + + α t α 0 z n + ρα 0 α 0 z n + α 0 t z n + β 0 t n z n + z n j F(z) z n a n z µ for 0 j n Then, we have α n t µ α n + α n t α n t + + α λ+t α λ t n λ + α λt α λ t n λ + + α t α 0 t n + ρα 0 α 0 t n + α 0 t t n + β 0 t n t n + z n j

5 AA Mogbademu et al / Journal of the Nigerian Mathematical Society xx (xxxx) xxx xxx 5 = z n a n z µ α n t + µ + α n α n + α n t + + α λ+ t n λ + + α λ t n λ α λ t n λ + α t n ρ α 0 t n ρ α 0 t n + α 0 + β 0 n = z n a n z µ ρ ( α 0 + α 0 ) t n + n α λ t n λ + µ α nt + α 0 α λ t n λ t n t n 4 5 If 6 α λ a n z µ > t n λ + µ α nt + α 0 n ie z µ > α λ a n a n t n λ + µ α nt + α 0 n, 9 then all the zeros of F(z) whose modulus is greater than or equal to t lie in 0 z µ α λ a n a n t n λ + µ α nt + α 0 n But those zeros of F(z) whose modulus is less than t already satisfy the above inequality and all the zeros of P(z) are also the zeros of F(z) Hence it follows that all the zeros of F(z) and hence of P(z) lie in z µ α λ a n a n t n λ + µ α nt + α 0 n 4 This completes the proof 5 As in Theorem, if all imaginary parts of the coefficients are also monotonic and non-negative, then we get the 6 following result Corollary 4 Let P(z) = n j=0 a j z j be a polynomial of degree n with complex coefficients If Re(a j ) = α j and I m(a j ) = β j, for 0 j n such that for some real t > 0, µ 0, λ, 0 λ n and ρ, 0 < ρ, 9 t n α n + µt n t n α n t λ+ α λ+ t λ α λ t λ α λ ρα 0, 0 and β n β n β β 0 > 0, z µ αλ a n a n t n λ + µ α nt + α 0 ρ( α 0 + a 0 ) + β n t n 4 Remark 5 If we set ρ =, Theorem reduces to Theorem For if we set µ = 0, Theorem reduces to the 5 following result 6

6 6 AA Mogbademu et al / Journal of the Nigerian Mathematical Society xx (xxxx) xxx xxx 4 5 Corollary 6 Let P(z) = n j=0 a j z j be a polynomial of degree n with complex coefficients If Re(a j ) = α j and I m(a j ) = β j, for 0 j n such that for some real t > 0, λ, 0 λ n and ρ, 0 < ρ, t n α n t n α n t λ+ α λ+ t λ α λ t λ α λ ρα 0, z a n α λ t n λ α nt + α 0 t n + n References [] Gauss KF Beitrage zur theorie der algebraischen Gleichungen Abh Ges Wiss Gottingen 50;4: 0 [] Cauchy AL Exercise de mathematique Oeuvres 9;(9): [] Govil NK, Rahman QI On the Eneström Kakeya theorem Tohoku Math J 96;0:6 [4] Joyal A, Labelle G, Rahman QI On the location of zeros of polynomials Canadian Math Bull 96;0:55 6 [5] Marden M Geometry of polynomials Math surveys, vol Providence, RI: Amer Math Soc; 949 [6] Milovanovic GV, Mitrinovic DS, Rassias ThM Topics in polynomials: Extremal properties, inequalities, zeros Singapore: World Scientific Publishing Co; 994 [] Aziz A, Zargar BA Some extensions of Eneström Kakeya theorem Glasňik Mathematiki 996;:9 44 [] Govil NK, Mctume GN Some extensions of Eneström Kakeya theorem Int J Appl Math 00;():45 5 [9] Aziz A, Zargar BA Bounds for the zeros of a polynomial with restricted coefficients Appl Math 0;:0 [0] Anderson N, Saff EB, Verga RS An extension of Eneström Kakeya theorem and its sharpness SIAM Math Anal 9;:0 [] Dewan KK, Govil NK On the Eneström Kakeya theorem J Approx Theory 94;4:9 44 [] Dewan KK, Govil NK On the Eneström Kakeya theorem J Math Anal Appl 99;0:9 6 [] Gardener RB, Govil NK Some generalizations of the Eneström Kakeya theorem Acta Math Hungar 99;4:5 4

Bounds for the Zeros of a Complex Polynomial with Restricted Coefficients

International Journal of Mathematics Research. ISSN 0976-5840 Volume 5, Number 2 (2013), pp. 277-282 International Research Publication House http://www.irphouse.com Bounds for the Zeros of a Complex Polynomial