A New Criterion for Meromorphic Multivalent Starlike Functions of Order γ defined by Dziok and Srivastava Operator

Transcription

1 Proceeding of the Paitan Academy of Science 5 : Coyright Paitan Academy of Science ISSN: rint online Paitan Academy of Science Reearch Article A New Criterion for Meromorhic Multivalent Starlie Function of Order defined by Dio and Srivatava Oerator Rabha M El-Ahwah Mohamed K Aouf Ali Shamandy and Shea M El-Deeb * Mathematic Deartment Faculty of Science Manoura Univerity New Damietta 3457 Egyt Mathematic Deartment Faculty of Science Manoura Univerity Manoura 3356 Egyt Abtract: In thi aer we introduce a ubcla M ; of meromorhic multivalent tarlie function of order defined by Dio and Srivatava oerator The main object of thi aer i to invetigate variou imortant roertie and characteritic for thi cla Further a roerty reerving integral i conidered Keyword and hare: Meromorhic function adamard roduct generalied hyergeometric function Mathematic Subject Claification 3C45 INTRODUCTION Let be the cla of function of the form: f a N {} which are analytic and -valent in the unctured unit dic U : and } U \{} For function f given by and g defined by g b N then the adamard roduct or convolution of f and g i given by f g ab g f 3 For real number q and β β β Z { }; j we now j define the generalied hyergeometric function F ; β β ; by ee for examle q q [5 9] F ; β β ; q q q β β! q ; N N {}; U 4 where θ i the Pochhammer ymbol defined in ν term of the Gamma function Γ by Γ θ ν ν ; θ C C \{} θ ν Γ θ θθ θ ν ν N; θ C Correonding to the function 5 Received May ; Acceted January 3 *Correonding author Shea M El-Deeb: heaeldeeb@yahoocom

2 78 Rabha M El-Ahwah et al h q ; β β ; defined by h ; β β ; q F ; β β ; q q we conider a linear oerator q; β β : which i defined by the following adamard roduct: ; β β f q h ; β β ; f q 6 7 We oberve that for a function f of the form we have ; β β f q Γ a where for convenience Γ! 8 q 9 β β If for convenience we write q ; β β then one can eaily verify from the definition 8 that ee [4] f f q q f q The linear oerator wa invetigated q recently by Liu and Srivatava [4] and Aouf [3] Some intereting ubclae of analytic function aociated with the generalied hyergeometric function were conidered recently by for examle Dio and Srivatava [6 7] Gangadharan et al [8] Liu [] We note that: i a; c f L a c f f a > c > ee Liu and Srivatava [3]; ii n ; f > N n D f f n n ee Aouf [] and Uralegaddi and Somanatha [6]; iii ν ν F ; f f ν ν > N ee Aouf [] Uralegaddi and Somanatha [6] and Yang [7] Maing ue of the oerator f we ay that a function f i in the cla ; M if it atifie the following inequality: Re q f f or equivalently to f Re f R and β β q R \ Z ; N; q N ; q ; ; U 3 We note the following intereting relationhi with ome of the ecial function clae which were invetigated recently: i M n ; M n n N [] ; ee Aouf ii M n ; M n n N ; ee Aouf and oen [4] Alo we note that: i M n ; M n; n > n D f n Re ; n D f n ii M a c; M a c; a c > 4

3 Meromorhic Multivalent Starlie Function of Order 79 L a c f Re L ac f 5 a a In thi aer along with other thing we hall how that a function f which atifie the condition i meromorhic multivalent tarlie in U More receiely it i roved that M ; of function in for the clae M ; M ; 6 hold If q β and then M ; i the cla of meromorhic multivalent tarlie function of order The tarliene of member M ; i a conequence of 6 of Further for µ > let µ F µ t µ f t dt 7 it i hown that F M q ; whenever f ; Alo it hown that if M M ; f then n n F t f t dt n 8 belong to ; M for F in U Some nown reult Bajaj [5] Goel and Sohi [] Ganigi and Uralegaddi [9] Aouf and oen [4] and Aouf [] are extended PROPERTIES OF TE CLASS M ; Unle otherwie mentioned we aume throughout thi aer that : q R and β β R \ Z N N q > ; U In roving our main reult we hall need the following lemma due to Jac [] Lemma Jac [] Suoe w be a noncontant analytic function in U with w If w attain it maximum value at aoint U on the circle r then w ζw where ζ i ome real number Theorem M ; M ; Proof Let f ; Then M q f Re f We have to how that imlie the inequality Re q f f Define w in U { : C and by q f q f 3 w w Clearly w i regular and w Equation 3 may be written a f f w w 4 Differentiating 4 logarithmically and uing the identity we obtain f f q q w w w 5 that i [ w ][ w ] q f f q

4 8 Rabha M El-Ahwah et al w w 6 w [ w ] w We claim that w in U For otherwie by Jac' Lemma there exit U uch that w ζw 7 where w and ζ From 6 and 7 we obtain q f f q w w [ w ] w Thu ζ w 8 q f Re q f > which contradict ence w in U and from 3 it follow that f ; M and in Theorem we obtain the following corollary Putting q n n > β N Corollary M n ; M n; Putting q a > and β c > in Theorem we obtain the following corollary Corollary M a c; M a c; Theorem Let f atify the condition q f Re q f c c Then µ µ > 9 F t f t dt µ µ belong to M ; Proof From the definition of F we have F µ f q q µ F q uing and the identity the condition 9 may be written a F q F µ Re F µ F µ µ We have to rove that imlie the inequality Re q F F Define w inu by F F q q w w Clearly w i regular and w The quation 3 may be written a F q F q w w 3 4

5 Meromorhic Multivalent Starlie Function of Order 8 Differentiating 4 logarithmically and uing the identity we obtain F F F F w [ ] w w The above equation may be written a F µ q q F µ q q q q F F F F w [ ] w w µ F F q q which by uing 3 and 4 reduce to F µ q q F µ q q F F 5 6 w w 7 w [ w ] µ µ w We claim that w in U For otherwie by Jac' Lemma there exit U uch that w ζw 8 where w and ζ From 7 and F µ q q F µ q F F q w w ζw [ ] w µ µ w Thu µ µ 9 q F µ q F Re q F µ q F > which contradict 9 ence w in U and from 3 it follow that F ; M Putting n n > q and β in Theorem we obtain the following corollary Corollary 3 Let f atify the condition n D f Re n D f n µ n µ then F i given by belong to M n; Putting q a > and β c > in Theorem we obtain the following corollary Corollary 4 Let f atify the condition L a c f Re L ac f

6 8 Rabha M El-Ahwah et al a µ a µ then F i given by belong to a c; M Theorem 3 If f ; then M n n F t f t dt n belong to ; Proof We have M for F in U µ f F q q µ F and q q q q 3 µ f F µ F 4 Taing µ in the above relation we obtain F F q q q q q f f which reduce to q F F F Thu q f f q q q F F 5 6 Re q q f Re q f from which it follow that 7 Re F F Then F ; Thi comlete the M roof of Theorem 3 Remar: i Taing q n n> β and in all our reult we obtain the reult obtained by Aouf []; ii Taing n n > q and β in all our reult we obtain the reult obtained by Aouf and oen [4]; iii Taing q n n> β and in all our reult we obtain the reult obtained by Ganigi and Uralegaddi [9] 3 REFERENCES Aouf MK New criteria for multivalent meromorhic tarlie function of order alha Proc Jaan Acad Ser A 69 no 3: Aouf MK A new criterion for meromorhic valent tarlie function Analele Stiintifice Ale Univ "ALICUZA" IASI 4: Aouf MK Certain ubclae of meromorhically multivalent function aociated with generalied hyergeometric function Comut Math Al 55: Aouf MK & M oen New criteria for meromorhic univalent function of order Nihonai Math J 5: Bajai SK A note on a cla of meromorhic univalent function Rev Roum Math Pure Al : Dio J & M Srivatava Clae of analytic function aociated with the generalied hyergeometric function Al MathComut 3: Dio J & M Srivatava Certain ubclae of analytic function aociated with the generalied hyergeometric function Integral Tranform Sec Funct 4: Gangadharan A TN Shanmugan & M Srivatava Generalied hyergeometric function with -uniformly convex function Comut

7 Meromorhic Multivalent Starlie Function of Order 83 Math Al 44 no : Ganigi MD & BA Uralegaddi New criteria for meromorhic univalent function Bull Math Soc Sci Math R S Roumanie N S 33 no : Goel RM & NS Sohi On a cla meromorhic function Gla Mat 7: I S Jac Function tarlie and convex of order J London Math Soc 3: Liu J-L Strongly tarlie function aociated with the Dio-Srivatava oerator Tamang J Math 35: no : Liu J-L & M Srivatava A linear oerator and aociated familie of meromorhically multivalent function J Math Anal Al 59: Liu J-L & M Srivatava Clae of meromorhically multivalent aociated with the generalied hyergeometric function Math Comut Modell 39: Srivatava M & PW Karlon Multile Gauian yergeometric Serie alted Pre Elli orwood Limited Chicheter John Wiley and Son New Yor Chicheter Bribane Toronto Uralegaddi BA & C Somanatha Certain clae of meromorhic multivalent function Tamang J Math 3: Yang D On a cla of meromorhic tarlie multivalent function Bull Int Math Acad Sinica 4: