Research Inventy: International Journal Of Engineering And ScienceVol.2, Issue 8 (March 2013), Pp 06-10Issn(e): 2278-4721,...
On The Zeros Of Analytic Functions…7Theorem 1: Let 0)(0 jjj zazf be analytic for tz  . If for some 0 ,......2210 ...
On The Zeros Of Analytic Functions…8II. LEMMAFor the proofs of the above theorems , we need the following lemma, which is ...
On The Zeros Of Analytic Functions…9222022200MaMtMataz ,we conclude that F(z) and therefore f(z) does not vanish in...
On The Zeros Of Analytic Functions…10if*0Mzttazi.e. iftazMzt 0* .Since the region defined bytazMzt 0* is precisel...
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  1. 1. Research Inventy: International Journal Of Engineering And ScienceVol.2, Issue 8 (March 2013), Pp 06-10Issn(e): 2278-4721, Issn(p):2319-6483, Www.Researchinventy.Com6On The Zeros of Analytic FunctionsM. H. GulzarP. G. Department of Mathematics University of Kashmir, Srinagar 190006Abstract : In this paper we consider a certain class of analytic functions whose coefficients are restricted tocertain conditions, and find some interesting zero-free regions for them. Our results generalize a number ofalready known results in this direction.Mathematics Subject Classification: 30C10, 30C15Key-words and phrases: Analytic Function , Coefficients, ZerosI. INTRODUCTION AND STATEMENT OF RESULTSRegarding the zeros of a class of analytic functions , whose coefficients are restricted to certainconditions, W. M. Shah and A. Liman [4] have proved the following results:Theorem A: Let 0)(0 jjj zazf be analytic in tz  . If for some 1k ,......2210  atatak .,and for some  ,,......,2,1,0,2arg  ja jthen f(z ) does not vanish in2222)1()1()1(kMMtkMtkz ,where10sin2)sin(cosjjj taakM .Theorem B: Let 0)(0 jjj zazf be analytic in tz  . If for some 1k and 0 ,............ 12210  kkktatatatak .,and for some  ,,......,2,1,0,2arg  ja jthen f(z ) does not vanish in22**22)1()1(*)1(kMtMkMtkz ,where100* sin2sincos)2(jjjkktaakktaaM .The aim of this paper is to generalize the above - mentioned results. In fact , we are going to prove thefollowing results:
  2. 2. On The Zeros Of Analytic Functions…7Theorem 1: Let 0)(0 jjj zazf be analytic for tz  . If for some 0 ,......2210  atata .,and for some real  and  ,,......,2,1,0,2arg  ja jthen f(z ) does not vanish in22202022200MaaMtMataz ,where1000 sin2)sin)(cos(jjj taaaaM.Remark1: Taking 0)1( ak  , 1k , Theorem 1 reduces to Theorem A.Also taking 0  , we get the following result, proved earlier by Aziz and Shah [2] :Corollary 1: Let 0)(0 jjj zazf be analytic for tz  such that for some 1k ,......2210  attaka .Then f(z) does not vanish in1212)1(kktktkz .Taking 0  and 0 , we get the following result proved earlier by Aziz and Mohammad [1] :Corollary 2: Let 0)(0 jjj zazf be analytic for tz  such that 0ja and,......3,2,1,01  jtaa jj .Then f(z) does not vanish in tz  .Theorem 2: Let 0)(0 jjj zazf be analytic for tz  . If for some 0 ,0 ,............ 112210   atatatata .,and for some real  and ,,......,2,1,0,2arg  ja jthen f(z ) does not vanish in22*2020*22*200MaatMMataz ,where10000* sin2sin1cos)12(jjj taaaataaM .Remark2: Taking 0)1( ak  , 1k , Theorem 2 reduces to Theorem B.The result of Aziz and Mohammad (Theorem 6 of [1]) follows from Theorem 2 by taking 0 .
  3. 3. On The Zeros Of Analytic Functions…8II. LEMMAFor the proofs of the above theorems , we need the following lemma, which is due to Govil andRahman [3]:Lemma : If2arg ja and for some t>0, ,,......,1,0,1 njata jj   , then]sin)(cos)[( 111    jjjjjj ataataata .III. Proofs Of The TheoremsProof of Theorem 1: Consider the function)()()( zftzzF ......)()(......)()(......))((2211002211002210ztaaztaaztaztaaztaatazazaatz)(0 zGzta   ,where 2110 )()()(jjjj ztaaztaazG  .Since ,......,2,1,0,2arg  ja j by using the lemma and the hypothesis, we have for tz  , sin}){(cos})[{()( 1010 taataatzG ......]}sin)(cos){( 2121   taataat.]sin2)sin)(cos1[(01000Mattaaaatjjj Since G(z) is analytic for tz  , G(0)=0, it follows by Schwarz’s lemma thatzMatzG 0)(  for tz  .Hence it follows that)()( 0 zGztazF  ][00 Mzttaza 0ifMzttaz0i.e. iftazMzt 0.Since the region defined bytazMzt 0is precisely the disk
  4. 4. On The Zeros Of Analytic Functions…9222022200MaMtMataz ,we conclude that F(z) and therefore f(z) does not vanish in the disk222022200MaMtMataz .That completes the proof of Theorem 1.Proof of Theorem 2: Consider the function)()()( zftzzF 1122110012211002210)(......)()(......)(......)()(......))((kkknnnztaaztaaztaaztaztaaztaaztaatazazaatz......)(...... 1  nnn zaa)(0 zGzta   .Since ,......,2,1,0,2arg  ja j by using the lemma and the hypothesis, we have for tz  ,1122110 )(......)()()(   ztaaztaaztaazG......)(...... 11  nnn zaa sin)(cos)[( 0101 ataatat ......]sin)(cos)(......sin)(cos)(sin)(cos)(......sin)(cos)(111111111111122122nnnnnnnnnnatatatatatatatatatatatatattaattaat100000sin2sin1cos)12(jjj taaaataaat *0 Mat ,where10000* sin2sin1cos)12(jjj taaaataaM .Since G(z) is analytic for tz  , G(0)=0, it follows by Schwarz’s lemma thatzMatzG *0)(  for tz  .Hence it follows that)()( 0 zGztazF  ][ *00 Mzttaza 0
  5. 5. On The Zeros Of Analytic Functions…10if*0Mzttazi.e. iftazMzt 0* .Since the region defined bytazMzt 0* is precisely the disk22*20*22*200MatMMataz ,we conclude that F(z) and therefore f(z) does not vanish in the disk22*20*22*200MatMMataz .That completes the proof of Theorem 2.REFERENCES[1] A. Aziz and Q. G. Mohammad, On the zeros of a certain class of polynomials and related analytic functions, J. Math. Anal.ppl.75(1980), 495-502.[2] A. Aziz and W. M. Shah, On the location of zeros of polynomials and related analytic functions, onlinear Studies 6 (1999), 91-101.[3] N. K. Govil and Q.I. Rahman, On the Enestrom-Kakeya Theorem II, Tohoku Math.J. 20 (1968), 126-136.[4] W. M. Shah and A. Liman, On Enestrom-Kakeya Theorem and related analytic functions, Proc. Indian Acad. Sci. (Math. Sci.), Vol.117, No.3, August 2007359-370.

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