The main aim of the present paper is to introduce a new class of multivalent analytic functions by using the familiar concept’s of convolution structure. The results investigated in the present paper include the characterization properties for this class of analytic functions. Some new and interesting consequences of our results are also pointed out

Agarwal, Ritu

Paliwal, G. S.

Parihar, Hari Singh

English

Scitech Research Journals

Journal of Progressive Research in Mathematics(JPRM)                                                                                                                                                           ISSN: 2395-0218      Volume 6, Issue 1 available at www.scitecresearch.com/journals/index.php/jprm                                741  SCITECH                          Volume 6, Issue 1  RESEARCH ORGANISATION|  Published online: December 17, 2015| Journal of Progressive Research in Mathematics www.scitecresearch.com/journals     Families of Multivalent Analytic Functions Associated with the Convolution Structure Hari Singh Parihar 1 , Ritu Agarwal 2  and G.S.Paliwal 3  1 Central University of Rajasthan, Kishangarh, Ajmer, INDIA-305817 Email:harisingh.p@rediffmail.com 2 Malaviya National Institute of Technology, Jaipur, INDIA-302017 Email: ragarwal.maths@mnit.ac.in 3 JECRC UDML College of Engineering, Kukas, Jaipur, INDIA-302028 Email:gaurishankarpaliwal@gmail.com     Abstract. The main aim of the present paper is to introduce a new class of multivalent analytic functions by using the familiar concept’s of convolution structure. The results investigated in the present paper include the characterization properties for this class of analytic functions. Some new and interesting consequences of our results are also pointed out. Keywords: Analytic functions; Convolution; Characterization properties; linear operators.  1  Introduction  Let pA  denote the class of functions that are analytic in the unit disk 1}|<|,:{= zCzzU   and consisting of the functions f  of the form   ),{1,2,3...}=,(,=)(1=Npzazzf nnpnp   (1.1)  where f  is analytic and p-valent in U . If pAf   is given by (1.1) and pAg  is given by   ){1,2,3...}=(,=)(1=Npzbzzg nnpnp   (1.2)  then the Hadamard product (or convolution) gf *  of f  and g , defined by   ).)(*(==))(*(1=zfgzbazzgf nnnpnp   (1.3)  In this article we study the class );(  gSp introduced in the following:  Journal of Progressive Research in Mathematics(JPRM)                                                                                                                                                           ISSN: 2395-0218      Volume 6, Issue 1 available at www.scitecresearch.com/journals/index.php/jprm                                742 Definition 1.1 For a given function pAzg )(  defined by (1.2), where 0nb , 1)(  pn , 1,2,...=p . We say that pAzf )(  is in ),;(  gSp provided that 0,))(*( zgf  and   )</{0};0;(>))(*()()*(1pCUzpzgfzgfzpRe   (1.4)  Note that )(=;1*11  SzzS  and ),(=;)(1 211  KzzS  are respectively, the familiar classes of starlike and convex functions of order   in U  (see for example,  [16]). Also, *=;01ppp SzzS   and ppp KzzS  =;0)(1 2 . For 1=p , the classes **1 =  SS  and  KK =1, where the classes *S  and K  stand essentially for the classes of starlike and convex functions of complex order, which were considered earlier by Nasr and Aouf  [11] and Wiatrowski  [17], respectively (see also  [9] and  [10]).  Remark: When   ,)!())(()())(()(=)(1=1=1=1=1= pnzpnBpnAzzgniisiiqiiiqiisipnp   where )1,...,=(),1,...,=( siCqiC ii    and the coefficients )1,...,=( qiRAi   and )1,...,=( siRBi   being so chosen that the coefficients nb  in (1.2) satisfying the following condition:   0,)!(1))(()())(()(=1=1=1=1= pnpnBpnAbiisiiqiiiqiisin (1.5)  then the class );(  gSp is transformed into a (presumably) new class ),,(  sqSp defined by     >))(][()()][(1:=),,(,',pzfLzfLzpReandAffsqSipsqipsqpp  (1.6)  {0}; CUz    and {0}).=,(0,1 01=1=  NNsqAB iqiisi The operator   )1,...,=(),(),...,;,...,;,...,(=)()( 111,, qizfBBLzfL ssqpsqipsq    involved in (1.6) is defined by the Chaurasia and Parihar (see for details  [3]). Special cases of the operator )()(, zfL ipsq   includes Dziok-Srivastava linear operator (cf.  [4, 5, 15]), Hohlov linear operator  [15], the Carlson-Shaffer linear operator  [2], the Ruscheweyh derivative operator  [14], the Barnardi-Libra -Livingston linear integral operator (cf.  [8, 7, 1]), and the Srivastava - Owa fractional derivative operators(cf.  [12, 13]) 2  Characterization Properties In this section, we establish two results, Theorem  2.1 and Theorem 2.3, which gives the sufficient Journal of Progressive Research in Mathematics(JPRM)                                                                                                                                                           ISSN: 2395-0218      Volume 6, Issue 1 available at www.scitecresearch.com/journals/index.php/jprm                                743 conditions for a function )(zf  defined by (1.1) and belongs to the class );()(  gSzfp . Theorem 2.1  Let pAzf )(  such that   );<(,<))(*()()*( 'Uzpppzgfzgfz   (2.1)  then );()(  gSzfp  provided that   ).<(0,|| ppp   (2.2)   Proof: In view of (2.1), we write   )()(=))(*()()*( 'zwppzgfzgfz  where 1|<)(| zw  for Uz . Now   ||1)(>)()()()(=)()(1=))(*()()*(1ppzwppzwReppzwppRepzgfzgfzpRe provided that pp|| . This completes the proof.  If we set ),<0{0};(||)(= pCpp    in Theorem  2.1, we obtain Corollary 2.2  If pAzf )(  such that   )<0{0};,(|,|)(<))(*()()*(pCUzppzgfzgfz  (2.3) then );()(  gSzfp .  Theorem 2.3. Let pAzf )(  satisfying the following inequality   ||)(|||]|)()[(1= pappnb nnpn (2.4)  )<{0};0});{1,2,3,...1,0(,( pCppnbUz n    then );()(  gSzfp .  Journal of Progressive Research in Mathematics(JPRM)                                                                                                                                                           ISSN: 2395-0218      Volume 6, Issue 1 available at www.scitecresearch.com/journals/index.php/jprm                                744 Proof: Suppose the inequality (2.4) holds true. Then in view of Corollary  2.2, we have   0||)(|||]|)()[(||||||)(||)(||)(||)()(=|))(*(|||)(|))(*()()*(|1=1=1=1=1='pappnbzabppapnbzbazpzapnbzgfpzgfpzgfznnpnpnnpnnnpnnnnpnpnnnpn This completes the proof.  Corollary 2.4  If pAzf )(  satisfying the following inequality   /{0};|,|||]||)[(1=CUzpappn npn  (2.5) then );()(  gSzfp .  Corollary 2.5  If pAzf )(  satisfying the following inequality   /{0};|,|||]||)[(1=CUzpappnn npn  (2.6) Then pKzf )( .  Corollary 2.6  If pAzf )(  satisfying the following inequality   ||)(||)!(1))(()())(()()](||)[(1=1=1=1=1= papnpnBpnAppn niisiiqiiiqiisipn (2.7)  )<0{0};;( pCUz    and {0}),=,(0,1 01=1=  NNsqAB iqiisi then ),,()(  sqSzfp .  References [1] Bernardi, S. D. (1969) Convex and starlike univalent functions, Trans. Am. Math. Soc., 135, 429-446. [2] Carlson, B.C. and Shaffer, D.B. (1984) Starlike and prestarlike hypergeometric functions, SIAM J. Math. Anal., 15(4), 737-745. [3] Chaurasia, V. B. L. and Parihar, H. S. (2010) On subordination for certain analytic functions associated with Fox-Wright psi functions, Bull. Belg. Math. Soc. Simon Stevin, 17(2), 1-7. 251-257. [4] Dziok, J. and Srivastava, H. M. (1999) Classes of analytic functions associated with the generalised hypergeometric function, Appl. Math. Comput., 103, 1-13. [5] Dziok, J. and Srivastava, H. M. (2003) Certain subclasses of analytic functions associated with the Journal of Progressive Research in Mathematics(JPRM)                                                                                                                                                           ISSN: 2395-0218      Volume 6, Issue 1 available at www.scitecresearch.com/journals/index.php/jprm                                745 generalised hypergeometric function, Integral Transform Spec. Funct., 14, 7-18. [6] Hohlov, Ju. E. (1978) Operators and operations on the class of univalent functions, Izv. Vyssh. Uchebn. Zaved. Mat., 10(197), 83-89. [7] Libera, R. J. (1965) Some classes of regular univalent functions, Proc. Am. Math. Soc., 17, 755-758. [8] Livingston, A. E. (1966) On the radius of univalence of certain analytic functions, Proc. Am. Math. Soc., 17, 352-357. [9] Murugusundaramoorthy, G., Srivastava, H.M. (2004) Neighborhood of certain classes of analytic functions of complex order, Journal of Inequalities in Pure and Applied Mathematics, 5(2), article(24). [10] Nasr, M. A. and Aouf, M. K. (1982) On convex functions of complex order, Mansoure Sci. Bull. Egypt, 9, 565-582. [11] Nasr, M. A. and Aouf, M. K. (1985) Starlike function of complex order, J. Natur. Sci. Math., 25(1), 1-12. [12] Owa, S. (1978) On the distortion theorems, I, Kyungpook Math. J., 18, 53-59. [13] Owa,S. and Srivastava, H. M. (1987) Univalent and starlike generalized hypergeometric functions, Canad. J. Math., 39, 1057-1077. [14] Ruscheweyh, S. (1975) New criteria for univalent functions, Proc. Amer. Math. Soc., 49,109-115. [15] Srivastava, H. M. (2001) Some families of fractional derivative and other linear derivative operators associated with analytic, univalent and multivalent functions, in: K. S. Lakshmi(Ed), Proceedings of International Conference on Analytic and Its Applications, Allied Publishers Limited, New Delhi and Chennai, 209-243. [16] Srivastava, H. M. and Owa, S.(Eds) (1992) Current Topic in Analytic Function Theory, World Scientific Publishing Company, Singapore, New Jersey, London and Hong Kong. [17] Waitrowski, P. (1971) Subordinating factor sequence for convex, Zeszyty Nauk. Uniw. Lodz. Nauki Mat. Przyrod. Ser. II, 39, 75-85.  

Families of Multivalent Analytic Functions Associated with the Convolution Structure

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Families of Multivalent Analytic Functions Associated with the Convolution Structure

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