AClassofMultivalentHarmonicFunctionsInvolvingSalageanOperator

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1 Global Journal of Science Frontier Research: F Matheatics Decision Sciences Volue 5 Issue Version 0 Year 05 Type : Double Blind Peer Reviewed International Research Journal Publisher: Global Journals Inc (USA Online ISSN: & Print ISSN: A Class of Multivalent Haronic Functions Involving Salagean Operator By Noohi Khan Aity University Lucknow, India Introduction- A continuous coplex valued function f=u+iv defined in a siply connected coplex doain D is said to be haronic in D if both u v are real haronic in D Let F G be analytic in D so that F(0=G(0=0, ReF = Ref=u, ReG = If=v by writing (F+iG/ = h, (FiG/ = g, The function f adits the representation f = h + g, where h g are analytic in D h is called the analytic part of f g, the co-analytic part of f GJSFR-F Classification : FOR Code : MSC 00: 30F5 AClassofMultivalentHaronicFunctionsInvolvingSalageanOperator Strictly as per the copliance regulations of : 05 Noohi Khan This is a research/review paper, distributed under the ters of the Creative Coons Attribution- Noncoercial 30 Unported License peritting all non coercial use, distribution, reproduction in any ediu, provided the original work is properly cited

2 Ref Ahuja,OP Jahangiri,JM, Multivalent Haronic starlike Functions, Ann Univ Mariae Curie Sklodowska, Section A, 55 ( (00, -3 A Class of Multivalent Haronic Functions Involving Salagean Operator Noohi Khan I Introduction A continuous coplex valued function f=u+iv defined in a siply connected coplex doain D is said to be haronic in D if both u v are real haronic in D Let F G be analytic in D so that F(0=G(0=0, ReF = Ref=u, ReG = If=v by writing (F+iG/ = h, (F-iG/ = g, The function f adits the representation f = h + g where h g are analytic in D h is called the analytic part of f g, the coanalytic part of f Ahuja Jahangiri [], [] introduce studied certain subclasses of the faily SH(, of all ultivalent haronic orientation preserving functions in = {: < } A function f in SH( can be expressed as f = h + g, where h g are analytic functions of the for For analytic function h ( S ( Salagean [3] introduced an operator D defined as follows: Whereas, Jahangiri et al [4] defined the Salagean operator Df( for ultivalent haronic function as follows: where, h ( a = + =, g( b (D h(' = D (D h( = D h( D f( = D h ( + ( D g( < 0 Dh( = h(, Dh( = D (h( = h'( n +, N = + a n + = + D h( a ( ( (3 Global Journal of Science Frontier Research F Volue XV Issue II V ersion I Year 05 8 Author: Departent of Matheatics Astronoy, Univesity of Lucknow, Lucknow 05 Global Journals Inc (US

3 A Class of Multivalent Haronic Functions Involving Salagean Operator Global Journal of Science Frontier Research F Volue XV Issue II V ersion I Year 05 8 In this paper we define a sub class H ( λ,, α of -valent haronic functions involving Salagean operator Df( as follows: Definition Let f ( = h( + g( be the haronic ultivalent function of the for (, then f H ( λ,, α if only if Df( Df( Re ( θ λ + λ > α (4 θ i where 0 α <, λ 0, = re θ D is defined by (3 f( We denote the subclass TH ( λ,, α consist of haronic functions in H ( λ,, α so that h g are of the for h ( a, = g ( = (, < Also note that TH ( λ,,o TH ( λ, The class H ( λ,, α provides a transition between two classes: Df( Df( Re as λ oves between 0 > α Re θ > α θ Denote H by by (0,, α P (, α H (,, α Q (, α In this paper first we obtained the sufficient coefficient condition for f( H ( λ,, α then it is shown that this coefficient condition is also necessary for f( TH Also distortion bounds, extree points, convex cobination, integral ( λ,, α operator, convolution condition, radius of convexity, radius of starlikeness for the functions f( TH ( λ,, α are obtained n + = D g( b D f( = i (D h(' ( (D g(' θ II Main Results a Theore (Sufficient coefficient condition for Η (λ,,α Assue that f = h + g, h g be given by ( λ 0, if λ + ( λ a + θ, = i f h + = g n + n + λ ( λ n,0 α < (6 then, f( H ( λ,, α 05 GlobalJournals Inc (US

4 A Class of Multivalent Haronic Functions Involving Salagean Operator b Reark The coefficient bound (6 in above theore is sharp for the function xn f( = + n + n + λ + ( λ (7 where α for λ 0 α <, then f H ( λ,, α Putting λ = 0 in Theore the following Corollary is obtained e Corollary 5 Let f = h + g be such that h g are given by ( let for 0 α <, then f P (, α Putting λ= in Theore the following Corollary is obtained f Corollary 6 Let f = h + g be such that h g are given by ( let for 0 α <, then f Q (, α + g Reark 7 H ( λ,, α H ( λ,, α for α α Also, Q (,α P (, α y n + n + λ ( λ n x n + y n = c Reark 3 For λ, n n n ( ( λ λ λ λ d Corollary 4 Let f = h + g be such that h g are given by ( let λ + ( λ a + n + n + λ ( λ n n + a + n + + a + n + n + n + n (8 (9 Global Journal of Science Frontier Research F Volue XV Issue II V ersion I Year Global Journals Inc (US

5 A Class of Multivalent Haronic Functions Involving Salagean Operator h Theore 8 (Coefficient inequality for ΤΗ (λ,,α Let f = h + g be so that h g are given by (5 Then, f TH ( λ,, α λ + ( λ a + n + n + λ ( λ (0 Global Journal of Science Frontier Research F Volue XV Issue II V ersion I Year where i Theore 9 (Distortion Bounds If f TH ( λ,, α λ, = r <, then + r ( ( λ f ( ( r ( ( + + λ + λ j Corollary 0 Let f TH ( λ,, α then for =r< λ k Theore (Extree Points Let f be given by (5 then f TH ( λ,, α ; λ if only if f ( = xn h n ( + yn g, (, where 0 α <, λ a = λ ( ( + + λ + λ + r ( ( f ( ( + r + ( + ( + λ ( (λ ( + ( + λ w : w < f ( + ( + ( + λ ( + ( + λ h ( =, h ( =, (n =,3 n + n + λ + ( λ g, ( = + (,(n =,,3 In particular, the extree points of TH ( λ,, α are {h n } {g } n + n + λ ( λ x 0, y 0, x = x y n n n, ( ( (3 (4 05 GlobalJournals Inc (US

6 A Class of Multivalent Haronic Functions Involving Salagean Operator l Theore (Convex Cobination If f i, (i =, belongs to TH ( λ,, α ; λ then the function tif i, ( is also in i= TH ( λ,, α where fi, is defined by i, n,i,i f = a + ( (i =, (5 i i i= 0 t <, t = Definition The haronic generalied Bernardi-Libera-Livingston integral operator for -valent functions is defined by c + c c + c c c c 0 0 L (f( = t h(tdt + t g(tdt, c > n Theore 3 (Integral Operator Let f TH ( λ,, α ; Thus L (D f ( belongs to the class TH ( λ,, α λ c o Theore 4 (Convolution Condition Let f TH ( λ,, α F TH ( λ,, α ; λ then the convolution p Theore 5 (Radius of Convexity The radius of convexity for the function q Theore 6 (Radius of Starlikeness The radius of starlikeness for the function n (f F ( = a A + n + ( B TH ( λ,, α f TH ( λ, is given by + r 0 =, for λ (λ f TH ( λ, + r 0 =, for λ (λ is given by L c(f( Global Journal of Science Frontier Research F Volue XV Issue II V ersion I Year References Références Referencias Ahuja,OP Jahangiri,JM, Multivalent Haronic starlike Functions, Ann Univ Mariae Curie Sklodowska, Section A, 55 ( (00, -3 Ahuja,OP Jahangiri,JM, Errata to Multivalent Haronic starlike Functions, Ann Univ Mariae Curie-Sklodowska, Vol LV, Sectio A 55 (00, -3, Ann Univ Mariae Curie Sklodowska, Sectio A, 56( (00, Global Journals Inc (US

7 A Class of Multivalent Haronic Functions Involving Salagean Operator 3 Salagean,GS, Subclass of Univalent Functions, Lecture in Math Springer Verlag 03(983, Jahangiri,JM, Murugusundaroorthy,G Vijaya,K, Salagean type Haronic Univalent Functions, South J Pure Appl Math, Issue (00, 77-8 Global Journal of Science Frontier Research F Volue XV Issue II V ersion I Year Global Journals Inc (US