A Certain Subclass of Multivalent Analytic Functions Defined by Fractional Calculus Operator

Transcription

1 British Journal of Mathematics & Comuter Science 4(3): SCIENCEDOMAIN international A Certain Subclass of Multivalent Analytic Functions Defined by Fractional Calculus Oerator Jamal M. Shenan * and Ghai S. Khammash Deartment of mathematics Alahar University-Gaa P. O. Box 77 Gaa Palestine. Deartment of mathematics Al-Aqsa University Gaa Stri Palestine. Original Research Article Received: 6 May 3 Acceted: Setember 3 Published: 3 November 3 Abstract In this aer we introduce a new subclass of multivalent analytic functions defined by fractional calculus oerator. Such results as subordination and suerordination roerties convolution roerties inequality roerties and other interesting roerties of this subclass are roved. Keywords: Analytic function multivalent function fractional differintegral oerator convex univalent function hadamard roduct subordination suerordination. Introduction Let H(U ) be the class of functions analytic in U { : C and } and H [ a ] the subclass of H(U ) with H H [] and H H [] consisting of functions of the form f () a a a.... Let A () denote the class of functions of the form n n n U f () a { 3...}; (.) which are analytic in the oen unit dis U and let A () A and A () A. * A function f A () of the form (.) is said to be in the class S (-valent) starlie of order if it satisfies the following inequality: be of multivalent *Corresonding author:

2 British Journal of Mathematics & Comuter Science 4(3) f f U. (.) Let f and F be members of H (U ) the function f is said to be subordinate to F or F is said to be suerordinate to f if there exists a function w analytic in U with w and w U such that f Fw. In such a case we write f F. In articular if F is univalent then f F if and only if f F and f U FU []. For two functions f () given by (.) and n g () b (.3) n n The hadmard roduct (or convolution) of f and g is defined by ( *)()( *)(). n f g a n bn g f (.4) n We recall the definitions of the fractional derivative and integral oerators introduced and studied by Saigo (cf.[6] and [] see also [89 and ]). Definition. Let and R of order of a function f is defined by function f then the generalied fractional integral oerator t I I f t F ; ; f t dt (.5) where the function f is analytic in a simly -connected region of the lane containing the origin and where the multilicity of t is removed by requiring log t to be real when t rovided further that f O for max. (.6) Definition. Let and R J of order of a function f is defined by then the generalied fractional derivative oerator 433

3 British Journal of Mathematics & Comuter Science 4(3) d t J f t F ; ; f t dt d (.7) n d J n f n n ; n N. d where the function f be analytic in a simly -connected region of the lane containing the origin with the order as given in (.6) and the multilicity of t is removed by requiring log t to be real when t. Note that I f D f (.8) J f D f (.9) where D f and D f integral and derivative oerator (cf. [3] and [4] see also [5]). are resectively the nown Riemann- Liouvill fractional Definition.3 For real number and the fractional oerator [5] U A A : and a ositive real number is defined in terms of n n ( ) n n n n n J and I U f a (.) which for f may be written as by U f J f ; I f ; (.) Where J f and I f are resectively the fractional derivative of f of order if and the fractional integral of f of order if. It is easily verified (see Choi [4] ) from (.) that 434

4 British Journal of Mathematics & Comuter Science 4(3) U f U f U f Note that (.). U f f (.3) ( ) ( ) The fractional differintegral oerator and f Mishra [5] and the fractional differential oerator Srivastava and Aouf [6]. We further observe that studied by Owa and Srivastava [4]. It is interesting to observe that ( ) is studied Patel and with ( ) was investigated by is the oerator introduced and U f f (.4) U f f (.5) By maing use of the differintegral oerator ( ) and the above mentioned rincile of subordination between analytic functions we introduce and investigate the following subclass of the class o A () f -valent analytic functions. Definition.4 A function f A () is said to be in the class S A B satisfies the following subordination condition: ( )( )( ) ( ) ; if it U f U f U f A (.6) U f B ( ; B A ; A B ; A ; ; and Re() ). It may be noted that for suitable choice of A B and the class S A B ; extends several classes of analytic and -valent functions studied by several authors such as Aouf and Seoudy [3] Shenan [4] Yang [7] Zhou and Owa [8] and Liu [6]. To rove our results we need the following definitions and lemmas. 435

5 British Journal of Mathematics & Comuter Science 4(3) Definition.5 ([]). Denote by Q the set of all functions f () that are analytic and injective on U /() E f where E () f { U: lim() f } and are such that f () for U /() E f. Lemma. ([]). Let the function h() be analytic and convex (univalent) in U with h(). Suose also that the function g () given by g c c (.7) ()... is analytic in U. If g () g ()()() ; ; h U (.8) Then g ()()()() q h t t dt h t and q() is the best dominant of (.8). Lemma. ([ 3]). Let q() be a convex univalent function in U and let \{} * with q q max. If the function g () is analytic in U and g ()()()() g q q then g ()() q and q() is the best dominant. Lemma.3 ([]). Let q() be convex univalent function in U and let. Further assume Re(). If g () H[()] q Q and g ()() g is univalent in U then 436

6 British Journal of Mathematics & Comuter Science 4(3) imlies g ()() q()()()() q g g q and q() is the best subordinate. Lemma.4 ([5]). Let the function F be analytic and convex in U. If f g A f g F then f ()( g F). Lemma.5 ([ 7]). Let f () a analytic and convex in U. If f ()() g a b. be analytic in U and then g () Lemma.6 ([8]). Let () g [ H ] and M g () L L where b and be M M ( ) (). If h() H[ ] satisfies the following subordination condition; g () ()() h M then Main Result (()) h (). U Theorem. Let f () S ; A B with (). Then 437

7 British Journal of Mathematics & Comuter Science 4(3) ( ) () U f () Au A q() u du (.) Bu B and q() is the best dominant. Proof. Define the function g () by ( ) U f g ()() U. (.) g is of the form (.7) and analytic in U. Differentiating (.) with resect to and Then () using (. ) we get ( )( )( ) ( ) U f U f U f g () A () g. (.3) U f () B Alying Lemma. to (.3) with () we get ( ) () () () Bt U f At q t dt () () Au A u du (.4) Bu B and q() is the best dominant. Theorem. Let q() be univalent function in U and let satisfies the following inequality: *. Suose also that q() q () max. q (.5) If f A satisfies the following subordination: 438

8 British Journal of Mathematics & Comuter Science 4(3) Then ( )( )( ) ( ) U f U f U f q() () q (.6) U f () ( ) U f and q() is the best dominant. q () Proof. Let the function g () (.3) and (.6) we find that be defined by (.). we now that (.3) holds true. Combining ()() ()() g g q q. (.7) ()() By using Lemma. and (.7) we easily get the assertion of Theorem.. Taing q() A in Theorem. we get the following result. B * Corollary. Let and B A. Suose also that If f A () B max. B satisfies the following subordination: ( )( )( ) () ( ) U f ()() B B U f U f U f A A B then ( ) U f A B and A B is the best dominant. 439

9 British Journal of Mathematics & Comuter Science 4(3) Theorem.3 Let () let q be convex univalent function in U and let with (). Also ( ) U f H [()] q Q and ( ) ( )( )( ) U f U f U f U f be univalent in U. If f ( ) ( )( ) ( ) q() U f U f q () () U f then ( ) U f q() and q() is the best subordinate. Proof. Let the function g () be defined by (.). Then ( ) q () ( )( )( ) q() U f U f U f () U f = g () g (). () By using Lemma.3 we easily get the assertion of theorem.3. Taing q() A in Theorem.3 we get the following result. B Corollary. Let q() be convex univalent function in U and B A (). Also let with 44

10 British Journal of Mathematics & Comuter Science 4(3) ( ) U f H [()] q Q and ( ) ( )( )( ) U f U f U f U f be univalent in U. If ( )( )( ) A A B ( ) B B U f () U f U f U f ()() Then ( ) A U f B and A B is the best subordinate. Combining the above results of subordination and suerordination we easily get the following ''sandwich-tye result''. Corollary.3 Let () q be convex function in U and let q () be univalent function in U let with (). let q () satisfy (.5). If ( ) U f H [()] q Q and ( ) ( )( )( ) U f U f U f U f is univalent in U and also 44

11 British Journal of Mathematics & Comuter Science 4(3) ( )( ) ( ) q () U f U f U f q () ( ) () U f q () q () () then ( ) U f ()() q q and () q and q () are resectively the best subordinate and dominant. Theorem.4 If and f () S ; R where f () S ; for ( ) then R ()(). (.8) The bound R is the best ossible. Proof. We begin by writing ( ) U f ()() g ; U. (.9) Then clearly the function g () is of the form (.7) is analytic and has a ositive real art in U. Differentiating (.9) with resect to and using the identity (.) we get ( )( )( ) U f U f U f ( ) U f g () g (). (.) () By maing use of the following well-nown estimate (see [9]): 44

12 British Journal of Mathematics & Comuter Science 4(3) g () r r (()) g r In (.) we obtain that ( )( )( ) U f U f U f ( ) U f r g (). () r It is seen that the right-hand side of (.) is ositive rovided that r (.8). R (.) where R is given by In order to show that the bound R is the best ossible we consider the function f ()() A defined by ( ) U f () ; U. Noting that ( )( )( ) U f U f U f ( ) U f. (.) () for R we conclude that the bound is the best ossible. Theorem.4 is thus roved. Theorem.5 Let f () S ; A B with (). Then () Aw n n n f () * () Bw (.3) n n n 443

13 British Journal of Mathematics & Comuter Science 4(3) where w () is an analytic function with w () and w () () U. with (). It follows from (.) that Proof. Let f () S ; A B ( ) U f () Aw () Bw (.4) where w () is an analytic function with w () and w () () U. By virtue of (.4) we easily find that ( ) () Aw U f () Bw Combining (.9) and (.5) we have. (.5) n n n () Aw *() f n () Bw n n (.6) The assertion (.3) of Theorem.5 can now easily be derived from (.6). Theorem.6 Let f () S ; A B with (). Then n n i n Be *() f n n n i Ae Proof. Let f () S ; A B imlies that ( U ; ). (.7) with (). We now that (.) holds true which ( ) U f Ae Be i i ( U ; ). (.8) It is easy to see that the condition (.8) can be written as follows: 444

14 British Journal of Mathematics & Comuter Science 4(3) ( ) i i U f Be Ae ( U ; ). (.9) Combining (.) and (.9) we easily get the convolution roerty (.7) asserted by Theorem.6. Theorem.7 Let and B B A A. Then ; ; S A B S A B. (.) Proof. Let f () S ; A B. Then ( )( )( ) ( ) U f U f U f A U f B B B A A we easily find that Since. ( )( )( ) ( ) U f U f U f A A (.) U f B B. Thus the assertion (.) holds for. If that is f () S ; A B by Theorem. and (.) we now that f () S ; A B ( ) U f A B At the same time we have that is (.) ( )( )( ) U f U f U f ( ) U f ( )( )( )( ) U f U f U f U f ( ).(.3) U f 445

15 British Journal of Mathematics & Comuter Science 4(3) A Moreover and the function B B A ; U is analytic and convex in U. Combining (.)-(.3) using Lemma.4 we find that ( )( )( ) ( ) U f U f U f A U f B that is f () S ; A B Theorem.8 Let f () S ; A B () () Au u du Bu which imlies that the assertion (.) of Theorem.7 holds. () and B A. Then with ( ) () () Bu U f Au u du The extremal function of (.4) is defined by. (.4) () ( ) () A u U f u du Bu Proof. Let f () S ; A B. (.5) with (). From Theorem. we now that (.) holds true which imlies that and ( ) () () su U Bu U f Au u du () () Au su u du Bu U () () Au u du Bu (.6) 446

16 British Journal of Mathematics & Comuter Science 4(3) ( ) () () inf U Bu U f Au u du () () Au inf u du Bu U () () Au u du Bu (.7) Combining (.6) and (.7) we get (.4). By noting that the function ( ) U f by (.5) belongs to the class S A B defined ; we obtain that equality (.4) is shar. The roof of Theorem.8 is evidently comleted. In view of Theorem.8 we easily derive the following distortion theorems for the class S ; A B. Corollary.4 Let f () S ; A B r we have () and B A. Then for with () () Aur r u du Bur () ( ) () Aur U f r u du. (.8) Bur The extremal function of (.8) is defined by (.5). By noting that ; From Theorem.8 we easily get the following results. Corollary.5 Let f () S ; A B with () () Au u du Bu. () and B A. Then 447

17 British Journal of Mathematics & Comuter Science 4(3) ( ) () () Bu U f Au u du. f defined by (.) be in the class S ; A B A B Theorem.9 Let () a n Then (.9) The inequality (.9) is shar with the extremal function defined by (.5). Proof. Combining (.) and (.6) we obtain ( ) ( )( )( ) U f U f U f U f a... A B () A... B (.3) An alication of Lemma.5 to (.3) yields a n A B. (.3) Thus from (.3) we easily arrive at (.9) asserted by Theorem.9. Theorem. Let and. with () P A P then If f () S ; A ( ) * f S ()(). Proof. Suose that f () S ; A. By (.6) we have 448

18 British Journal of Mathematics & Comuter Science 4(3) ( ) ( )( )( ) U f U f U f A. (.3) U f Let the function g () be defined by (.). We then find from (.) and (.3) that ()() () g () At t dt We now suose that () (). ( ) ( ) U f ()() h ; ; U. (.33) U f Then h H [ ]. It follows from (.3) and (.33) that g () [()() h] A U. (.34) An alication of Lemma.6 to (.34) yields (()) h U. (.35) Combining (.33) and (.35) we find that ( ) U f ()(()) h ; ( ) ;. U U f (.36) The assertion of Theorem. can now easily be derived from (.) and (.36). Acnowledgements The authors would lie to exress many thans to the referees for their valuable suggestions. Cometing Interests Authors have declared that no cometing exists. 449

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21 British Journal of Mathematics & Comuter Science 4(3) [8] Zhou ZZ Owa S. Convolution roerties of a class of bounded analytic functions Bull. Austral. Math. Soc. 99;45: Shenan and Khammash; This is an Oen Access article distributed under the terms of the Creative Commons Attribution License ( htt://creativecommons.org/licenses/by/3.) which ermits unrestricted use distribution and reroduction in any medium rovided the original wor is roerly cited. Peer-review history: The eer review history for this aer can be accessed here (Please coy aste the total lin in your browser address bar) 45