 Applied Mathematics, 2010, 3, 555-560 doi:10.4236/am.2010.16073 Published Online December 2010 (http://www.SciRP.org/journal/am) Copyright © 2010 SciRes. AM On The Eneström-Kakeya Theore m Gulshan Singh1, Wali Mohammad Shah2 1Bharat hiar Univers ity, Coimbatore, India 2Department of Mat hem at ic s, Kashmir University, Srinagar, India E-mail: gulshansingh1@rediffmail.com , wmshah@rediffmail.com Received August 9, 2010; revised November 17, 2010; accepted November 20, 2010 Abstract In this paper, we prove some generalizations of results concerning the Eneström-Kakeya theorem. The re-sults obtained considerably improve the bounds by relaxing the hypothesis in some cases. Keywords: Polynomial, Zeros, Eneström-Kakeya theorem 1. Introduction and Statement of Results The following result due to Eneström and Kakeya  is well known in the theory of distribution of the zeros of polynomials. Theorem A. If =0:= njjjPz az is a polynomial of degree n such that 12 10,nn naa aaao then Pz does not vanish in >1z In the literature, [2-8], there exist extensions and ge-neralizations of Eneström-Kakeya theorem. Joyal, La-belle and Rahman  extended this theorem to a poly-nomial whose coefficients are monotonic but not neces-sarily non negative by proving the following result. Theorem B. Let =0:= njjjPz az be a polynomial of degree n such that 12 10,nn naa aaa then all the zeros of ()Pz lie in 001.nnzaaaa Dewan and Bidkham  generalized Theorem B and proved the following: Theorem C. Let =0:= njjjPz az be a polynomial of degree n such that for some >0t and 0<,n 111110,nnnnata tatattaa then Pz has all the zeros in the circle 0021.nnnnatzaaaatt By using Schwarz's Lemma, Aziz and Mohammad  generalized Eneström-Kakeya theorem in a different way and proved the following: Theorem D. Let =0:= njjjPz az be a polynomial of degree n with real positive coefficients. If 12>0tt can be found such that 121 1220rr rattatta , for =1,2, ,1rn 11==0,naa then all the zeros of Pz lie in 1zt Aziz and Zargar  also relaxed the hypothesis of Eneström-Kakeya theorem in a different way and proved the following result. Theorem E. Let=0:= njjjPz az be a polynomial of degree n such that for some 1K, 12 10>0,nn nKaaaa a then all the zeros of Pz lie in 1.zK K While studying Theorem E, a natural question arises that what happens if we relax the hypothesis of Theorem D in a similar way and only assume that 121 1220rr rattatta , for =2,3,,rn In this paper, we study such a case and prove a more general result from which many known results follow on a fairly uniform procedure. Infact we prove: Theorem 1. Let =0:= njjjPz zbe a polynomial of degree n such that =jjjaib where ja and jb, 0,1 ,jn are real numbers and if 12>0tt can be found such that for =2,3, ,rn G. SINGH ET AL. Copyright © 2010 SciRes. AM 556 121 1220,rr rattat ta 121 1220,rr rbttbttb and for some 1K, 0,)( 121 nn attKa 0,)( 121 nn bttKb then all the zeros of Pz lie in 121zK tt  ,R where 21221 1111=nn nnnntRKabttabt abt  001120 121120 1211111nnabattatt bttbtttt  2001.ntabt  The following interesting result immediately follows from Theorem 1, if we assume that all the coefficients of the polynomial Pz are real. Corollary 1. Let =0:= njjjPz az be a polynomial of degree n with real coefficients. If 12>0tt can be found such that 121 1220,rr rattat ta for =2,3, ,rn and for some 1K, 12 10,nnKatta   then all the zeros of Pz lie in 121zK tt  *,R where *2122 10111111=nnnnntRKattataaatt  21120120111.nntatta ttatt  Remark 1. If we assume that all the coefficients of Pz are real and positive, then for =1K, Corollary 1 satisfies the statement of Theorem D and a simple calcu-lation shows that in this case also all the zeros of Pz lie in 1.zt Next, if in the Theorem 1, we take 2=0t and assume that coefficients to be real, we get the following: Corollary 2. Let =0:= njjjPz az be a polynomial of degree n with real coefficients. If for some >0t and 1K, 1212 10>,nn nnn nKatatatata  then all the zeros of Pz lie in 001.nnnnaatzK tKaatt  Remark 2. If we put 1=t in Corollary 2, we get the result due to Aziz and Zargar  and for 1=t, 1=K, Corollary 2 reduces to Theorem B. We next prove the following more general result which is of independent interest. Theorem 2. Let=0:= njjjPz z be a polynomial of degree n such that jjj iba = where ja and jb, 0,1,2, ,jn are real numbers. If 0>21 tt can be found such that for 2,3, ,1rn 1211220,rr rattat ta  121 1220,rr rbttbttb  and for some real numbers u and v , 11, vu 12 10,nnuatta  12 10,nnvbttb  then all the zeros of )(zP lie in 12 11,nnnua ivbztt R  where 2112211111=nn nnnntRuavbttabtabt  0011201 211201211111nnabattatt bttbtttt  2001}.ntabt If in Theorem 2, we take 112=nnauat t and 112=,nnbvbt t so that 1,1,  vu we get the following: Corolla ry 3 . Let =0:= njjjPz z be a polynomial of degree n such that jjj iba = where ja and jb, 0,1,2, ,jn are real numbers. If 0>21 tt can be found such that 0)( 221121  rrrattatta , for r = 2,3,,n 121122() 0rr rattat ta , for r = n+1 0)( 221121 rrrbttbttb , for r = 2,3,,n G. SINGH ET AL. Copyright © 2010 SciRes. AM 557121 122() 0rr rbttbt tb, for r = n +1, then all the zeros of )(zP lie in *112 1,nnzttR where  *2121111111=nnn nnntRabtababt  0 0112012112 01211111nnabattatt bttbtttt 200111.nntabtt  In particular, if 1211220rr rattat ta, for 1, 2,,rn 121 1220rr rattat ta, for 1rn, 121 1220rr rbttbttb, for 1, 2,,rn 121 1220rr rbttbttb, for 1rn, then 1120 120,atta tt 1120 120bttb tt and we get in this case all the zeros of Pz lie in  11221 11.nnnn nnnzttabtab Remark 3. A result of Shah and Liman [7, Theorem 1] is a special case of Corollary 3, if we assume that all the coefficients of Pz are real. The following result also follows from Theorem 2, if we assume that 0=2t and 1.=1t] Corollary 4. Let =0:= njjjPz z be a polynomial of degree n such that jjj iba = where ja and jb, 0,1,2, ,jn are real numbers. If for some 1uand 1v, 100nnua aa, 100nnvb bb , then all the zeros of Pz lie in 1nn nnnnua ivbua vbz Many other known results and generalizations simi-larly follows from Theorem 2 with suitable substitutions. We leave this to the readers. 2. Proofs of the Theorems Proof of Theorem 1. Consider the polynomial 21=fztztzPz 2112112 1122=nn nnn n nnnzttzttttz     22121 1201120 12012ttttzttttztt  (1) 21 11212112 1122=1nn nnnn nnnnnzK ttzKttz ttttz    22121 1201120 12012ttttzttttztt  (2)  21 11212112 1122=1 ...nn nnnn nnnnnzKttz Kattaz attattaz   2 121211 2011201 20121 21nnnattattazattattzattiKbttb z  21211 2221211 2011201 2012(())nnn nbttbttbzbttbttbzbttbtt zbtt . This gives    111212112 11221nnnnnnnnnfzzzKttKattazattattaz  2 12121120112012012121nnnattattazattattzattKbttbz   21211 2221211 20112012012nnn nbttbtt bzbttbtt bzbttbttzbtt  . G. SINGH ET AL. Copyright © 2010 SciRes. AM 558 112121121121122=1nnnn nnnnnzzKttKattaKbttb attatta  121 12221211202121 120111nn nnbttbtt battattabttbtt bzz   112012112 012012012111nnattattbttbttattbttzz . For 1>zt, we have by using hypothesis    11212112 11nnnn nnfzzzKt tKat taKbt tb  1211 2212112211nnnnn nattat tabttbt tbt    21211202121120111nattat tabttbt tbt   1120 121120 1201201211111>0nnattattbttbttattbtttt , if   002212121222 111 1111 1111>nnnnnnnnnabttz KttKattKbttatbtabt ttt   221120 121120 12001111nnnttattatt bttbttabttt . Therefore, for 1zt, >0,fz if  2121221 1001111111> nn nnnnntz KttKabttabtababtt   21120 121120 1200111nntattattbttbtta btt . Hence all the zeros of fz whose modulus is greater than 1t lie in the circle  2121221 1001111111nnnnnnntzKttKabttabtababtt   21120 121120 1200111nntattattbttbtta btt . Since all the zeros whose modulus is less than 1t already lie in this circle, we conclude that all the zeros of fz and therefore Pz lies in  2121221 1001111111nnnnnnntzKttKabttabtababtt   21120 121120 1200111nntattattbttbtta btt . This completes the proof of the Theorem 1. Proof of Theorem 2. Consider the polynomial G. SINGH ET AL. Copyright © 2010 SciRes. AM 55921=fztztzPz 2112112 1122=...nn nnn n nnnzttzttttz    22121 120112012012ttttzttttztt  21 2121121122212 1120= ...nn nnn n nnnzatt azattatt a zattatt az  111201 20121211211 22nnnnnn nattattzattibtt bzbttbtt bz  22121 120112012012bttbtt bzbttbttzbtt 21 112121121122=1nn nnnn nnnnnzuattzuatt azattattaz  211212 1120112012012121211nnnnnattattazatt att zattivbttzvbttbz   21211 2221211 2011201 2012...nnn nbttbtt b zbttbtt bzbttbttzbtt   21112121121122=nnnnnnnn nnnnnzuaivbttzuattazattattaz  2 121211201120120121 21(()) nnnattat tazattat tzattivbt tbz  21211 22212112011201 2012nnn nbttbtt b zbttbttbzbttbttzbtt . This gives   1 11212112 11221nnnnnnnnnnnnua ivbfzzzttuatt azattattaz    2 121211 2011201 20121 21nnnattattazattattzattvbttb z   2121 1222121120112012012nnn nbttbtt bzbttbtt bzbttbttzbtt .  11212112 1=1nnnnnnnnnua ivbzzttuatt avbtt b  1211 2212112211201211201211nnnnn nnattattabttb ttbattattbttbttzz  012 01211nattbtt z  For 1>zt, we have   1121211211nnn nnn nnnua ivbfzzzttuat tavbt tb    1211 2212112211201 211201 21111nnnnn nnatta ttabttbttbattattbttbtttt  012 012111nattbtt t . By using hypothesis, this gives  12121221 100111111nnn nnn nnnnnua ivbtfzzzttuavbttabtababtt  G. SINGH ET AL. Copyright © 2010 SciRes. AM 560  2112012112 01200111>0,nntattattbttbtta btt  if 2121221 1001111111>nn nn nnnnnnua ivbtzttuavbttabtababtt  21120 121120 1200111nntattattbttbttabtt . Hence all the zeros of fz whose modulus is greater than 1t lie in the circle 21212211001111111nn nn nnnnnnua ivbtzttuavbttabt ababtt   21120 121120 1200111nntattattbttbttabtt . Since all the zeros whose modulus is less than 1t already lie in this circle, we conclude that all the zeros of fz and therefore Pz lies in 12 11,nnnua ivbztt R  where  211221 1001120121120121111 1111=nn nnnn nntRuavbttabt ababattattbttbtttt t 2001ntabt . This proves Theorem 2 completely. 3. Acknowledgements The authors are grateful to the referee for useful sugges- tions. 4. References  M. 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