ON CERTAIN SUBCLASS OF p-valent FUNCTIONS WITH POSITIVE COEFFICIENTS (Berkenaan Subkelas Fungsi p-valen Tertentu Berpekali Positif)

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1 Joual of Quality Measuemet ad Aalysis Jual Peguua Kualiti da Aalisis JQMA 10(2) 2014, ON CERTAIN SUBCLASS OF -VALENT FUNCTIONS WITH POSITIVE COEFFICIENTS (Beeaa Subelas Fugsi -Vale Tetetu Beeali Positif) ANAS ALJARAH & MASLINA DARUS ABSTRACT I this aticle, a cetai diffeetial oeato Sα,β,λ,δ, f( ) ad a subclass S,( α,β,δ,λ, ) fo aalytic fuctios with ositive coefficiets of the fom f ( ) = + a = + 1 ae itoduced. Some oeties such as the coefficiets iequalities, distotio theoem, adii of stalieess ad covexity, closue theoems ad exteme oits ae give. A class esevig a itegal oeato is also stated. Keywods: aalytic fuctio; stalie fuctios; covex fuctios; itegal oeato ABSTRAK Dalam maalah ii, egoeasi embea tetetu Sα,β,λ,δ, f( ) da subelas S,( α,β,δ,λ, ) bagi fugsi aalisis beeali ositif dalam betu f ( ) = + a = + 1 dieeala. Bebeaa sifat seeti etasamaa eali, teoem eota, jejai ebabitaga da ecembuga, teoem tutua da titi estem dibei. Kelas megeala egoeasi amia juga diyataa. Kata uci: fugsi aalisis; fugsi babitag; fugsi cembug; egoeasi amia 1. Itoductio ad Pelimiaies Let A ( ) deote the class of fuctios of the fom: ( ) = + a f ( U,! := {1,2,3,...}), (1.1) which ae aalytic ad -valet i the oe uit disc U = {! : < 1}. A fuctio f( ) i A ( ) is said to be -valet stalie of ode γ if ad oly if Re f ' () f () > γ ( U ), (1.2) fo some γ(0 γ ). We deote by S *(γ ) the class of all -valet stalie fuctios of ode γ. Similaly, a fuctio f( ) i A ( ) is said to be -valet covex of ode γ if ad oly if Re 1+ f '" () f () > γ ( U ), (1.3) fo some γ(0 γ < ). We deote by C(γ) the class of all -valet covex of ode γ.

2 Aas Aljaah & Maslia Daus Fo the fuctio f A( ), we defie the followig ew diffeetial oeato: S 0α,β,λ,δ, f () = f (), S 1α,β,λ,δ, f () = [1 ( λ δ )( β α )] f () + [( λ δ )( β α )]f ' (),, ad ( Sα,β,λ,δ, f () = S Sα 1 f (),β,λ,δ, = + ) ((λ δ )(β α )( ) + 1) (1.4) a, {} fo f A( ),α, β,δ, λ 0, λ > δ, β > α,!, = 1,2,3,... ad! =! 0. It is easily veified fom (1.4) that ( λ δ )( β α )(Sα,β,λ,δ, f ())' = Sα+1 f () (1 ( λ δ )( β α ))Sα,β,λ,δ, f ().,β,λ,δ, (1.5) Rema 1: Whe = 1, we have the oeato itoduced ad studied by Ramada ad Daus (2011). Maig use of the oeato Sα,β,λ,δ, f ( ) we say that a fuctio f ( ) A( ) is i the class S, (α, β,δ, λ, ) if it satisfies the followig iequality (Aouf 2012): { ( ℜ S f () α +1 S f () ) } > β, ' (1.6) fo some α (α > 0), β (0 β < ),!,! 0 ad fo all U. We ote that (Altitas et al. 1994), M 0 (1,α, β ) = M (1,α, β ). (1.7) I Ualeggadi ad Gaigi (1986), they defied the class H * (α) as follows: { f () A( ) : ℜ { +1 } } f '() > α,0 α <, U. (1.8) Some othe subclasses of the class A( ) wee studied (fo examle) by Cho et al. (1987, 1989), Liu (2000), Liu ad Sivastava (2001; 2004), Joshi et al. (1995), Raia ad Sivastava (2006), Aouf ad Shammay (2005). I this ae, coefficiet iequalities, distotio theoem, adii of stalieess ad covexity, closue theoems, exteme oits ad the class esevig itegal oeatos of the fom F() = c + c 1 t f (t)dt fo all c >, c 0 fo the class S, (α, β,δ, λ, ) ae cosideed. 42

3 O cetai subclass of -valet fuctios with ositive coefficiets 2. Coefficiet Iequalities I this sectio, we give a ecessay ad sufficiet coditio fo a fuctio f ( ) give by (1.1) to be i the class S, (α, β,δ, λ, ). Theoem 2.1. A fuctio f ( ) give by (1.1) is i S, (α, β,δ, λ, ), whee α > 0, β,δ, λ 0,,! ad! 0 if ad oly if ((λ δ )(β α )( )) (α 1)a (1 α β ). (2.1) Poof. Suose that f ( ) S, (α,β,δ,λ, ). The we fid fom (1.6) that + (( λ δ )( β α )( ) + 1) a ℜ >β +1 1 α (( λ δ )( β α )( ) + 1) a ℜ (1 α ) 2 (( λ δ )( β α )( )) (α 1)a + > β (α > 0, β,δ,λ 0,! ad! 0 ). If we choose = < 1, the we have 1 α ((λ δ )(β α )( )) (α 1)a β fo α > 0, β,δ,λ 0,! ad! 0, which is equivalet to (2.1). Covesely, let us suose that the equivalet (2.1) holds tue. The we have S f () α +1 (S f ()) (1 α ) 2 = (( λ δ )( β α )( )) (α 1)a + ((λ δ )(β α )( )) (α 1)a + 1 α β fo α > 0, β,δ,λ 0,! ad! 0, which imlies that f () S, (α, β,δ, λ, ). This gives the equied coditio. Hece the theoem follows. Coollay 2.2. Let the fuctio f ( ) be defied by (1.1). If f S, (α,β,δ,λ, ). The 43

4 Aas Aljaah & Maslia Daus a (1 α β ) (( λ δ )( β α )( )) (α 1) ( ;!;! 0 ). (2.2) The equality i (2.2) is attaied fo the fuctios f ( ) give by f ( ) = + (1- α - β) ((λ - δ)(β - α)( - )) (α - 1) ( ; ; 0 ). (2.3) 3. Distotio Theoem A distotio oety fo fuctios i the class S, (α,β,δ,λ, ) is cotaied i the followig theoem. Theoem 3.1. Let the fuctios f ( ) give by (1.1) be i the class S, (α,β,δ,λ, ), whee α > 0, β,δ,λ 0, ad 0. The fo = < 1, we have (1 α β) ((λ δ)(β α)( )) (α 1) f () + 1+ (1 α β) ((λ δ)(β α)( )) (α 1) 1+ ad (1 α β)(1+ ) ((λ δ)(β α)( )) (α 1) (1 α β)(1+ ) f () +. ((λ δ)(β α)( )) (α 1) Poof. Sice f S, (α,β,δ,λ, ), fom Theoem 2.1 eadily yields the iequality a (1 α β) ((λ δ)(β α)( )) (α 1). Thus, fo = < 1, ad maig use of (3.1) we have f () + a a (1 α β) ((λ δ)(β α)( )) (α 1) ad f () a a (1 α β) ((λ δ)(β α)( )) (α 1).

5 O cetai subclass of -valet fuctios with ositive coefficiets Also fom Theoem 2.1, it follows that ((λ δ)(β α)( )) (α 1) a 1+ ((λ δ)(β α)( )) (α 1)a (3.2) (1 α β). Hece f () a (1 α β)(1+ ) ((λ δ)(β α)( )) (α 1) a ad f () 1 a 1 1 (1 α β)(1+ ) ((λ δ)(β α)( )) (α 1) a. This comletes the oof of Theoem Radii of Stalieess ad Covexity I this sectio, we detemie the adii of stalieess ad covexity fo fuctios i the class S, (α,β,δ,λ, ). The esult is give as follows. Theoem 4.1. Let the fuctio f ( ) defied by (1.1) is i the class S, (α,β,δ,λ, ), whee α > 0, β,δ,λ 0, ad 0. The f ( ) is stalie of ode δ(0 δ < 1) i < 1, whee 1 ( δ)( δ)((λ δ)(β α)( )) (α 1) = if 1 +1 (1 + δ)(1 α β) ( +1). (4.1) The esult is sha fo the fuctio f ( ) give by (2.3). Poof. It suffices to ove that f ( ) - - δ, f ( ) (4.2) fo < 1. We have 45

6 Aas Aljaah & Maslia Daus f () f () = + (1 )a a (1 )a. (4.3) 1+ a Hece (4.3) holds tue if o (1 )a ( δ) 1+ a (1 + δ) a 1 ( δ). With the aid of (2.1), (4.4) is tue if (4.4) (1 +δ) ( δ) ((λ δ)(β α)( )) (α 1) (1 α β). (4.5) Solvig (4.5) fo, we obtai ( δ)( δ)((λ δ)(β α)( )) (α 1) (1 +δ)(1 α β) 1 ( +1). This comletes the oof of Theoem 4.1. Theoem 4.2. Let the fuctio f( ) defied by (1.1) is i the class S,( α,β,δ,λ, ), whee α >0, β,δ,λ 0, ad 0. The f( ) is covex of ode δ( 0 δ <1) i < 2, whee 2 = if +1 (1 δ)((λ δ)(β α)( )) (α 1) (1 +δ)(1 α β) 1 1 ( +1). (4.6) The esult is sha fo the fuctios f( ) give by (2.3). Poof. By usig the same techique emloyed i the oof of Theoem 4.1, we ca show that f ' " ( ) f ( ) - δ, (4.7) fo < 2, with the aid of Theoem 2.1. Thus, we have the assetio of Theoem

7 O cetai subclass of -valet fuctios with ositive coefficiets 5. Closue Theoems Let the fuctios f j ( ), j = 1,2,..., l be defied by f j ()= + a,j ( a,j 0), (5.1) fo U. I this sectio, we ove the closue theoem fo the class S,( α,β,δ,λ, ). The esults ae give i the followig theoems. Theoem 5.1. Let the fuctios f j( ) defied by (5.1) be i the class S,( α,β,δ,λ, ), whee α >0, β,δ,λ 0, ad 0 fo evey j = 12,,...,l. The the fuctio Gdefied ( ) by G()= + b ( b 0) (5.2) is a membe of the class S,( α,β,δ,λ, ), whee b = 1 l l a,j j=1 ( +1). Poof. Sice f j( ) S,( α,β,δ,λ, ), it follows fom Theoem 2.1 that ((λ δ)(β α)( )) (α 1) a (1 α β). fo evey j = 12,,...,l. Hece, = = 1 l ((λ δ)(β α)( )) (α 1) b l j=1 l 1 l ((λ δ)(β α)( )) (α 1) a,j l j=1 ((λ δ)(β α)( )) (α 1) a,j 1 (1 α β) = (1 α β), l j=1 which (i view of Theoem 2.1) imlies that G ( ) S,( α,β,δ,λ, ). Theoem 5.2. The class S,( α,β,δ,λ, ) is closed ude covex liea combiatio, whee α >0, β,δ,λ 0, ad 0. Poof. f ()= j + a,j ( a,j 0; j =1,2;!) (5.3) 47

8 Aas Aljaah & Maslia Daus be i the class S,(α,β,δ,λ, ). It is sufficiet to show that the fuctio h() defied by h() = t f 1()+ (1 t) f 2() (0 t 1) (5.4) is also i the class S, (α,β,δ,λ, ). Sice fo 0 t 1, h() = + t a,1 + (1 t) a,2 (0 t 1), (5.5) we obseve that ((λ δ)(β α)( )) (α 1) t a =t,1 + (1 t) a,2 ((λ δ)(β α)( )) (α 1)a,1 + (1 t) ((λ δ)(β α)( )) (α 1) a,2 t(1 α β) + (1 t)(1 α β) = (1 α β). Hece h( ) S, (α,β,δ,λ, ). This comletes the oof of Theoem Itegal Oeatos I this sectio, we coside the itegal tasfoms of fuctios i the class S, (α,β,δ,λ, ). Theoem 6.1. If the fuctio f ( ) give by (1.1) is i the class S, (α,β,δ,λ, ), whee α > 0, β,δ,λ 0, ad 0 ad let c be a eal umbe such that c > -, the the fuctio F ( ) defied by F() = c + c 1 f (t) dt c t 0 (6.1) is also belogs to the class S, (α,β,δ,λ, ). c+ Poof. Fom (6.1), it follows that F() = + b, whee b = a. Theefoe + c ((λ δ)(β α)( )) (α 1)b = c+ ((λ δ)(β α)( )) (α 1) + c a ((λ δ)(β α)( )) (α 1)a (1 α β), sice f () S,(α,β,δ,λ, ). Hece by Theoem 2.1, F() S,(α,β,δ,λ, ). 48

9 O cetai subclass of -valet fuctios with ositive coefficiets 7. Exteme Poits Now, we detemie exteme oits fo fuctios i the class S, (α,β,δ,λ, ). Theoem 7.1. Let f ( ) = ad f = + (1 α β) ((λ δ)(β α)( )) (α 1). The f S, (α,β,δ,λ, ) if ad oly if it ca be exessed i the fom f () = ξ f ()+ ξ (), U, whee ξ 0 ad ξ = 1 f ξ. Poof. Suose that f () = ξ f ()+ ξ f () (1 α β) f () = 1 ξ + ξ + ((λ δ)(β α)( )) (α 1) = + (1 α β) ξ ((λ δ)(β α)( )) (α 1). The fom Theoem 2.1 we have (1 α β) ((λ δ)(β α)( )) (α 1) ((λ δ)(β α)( )) (α 1) ξ (1 α β). Hece f S, (α,β,δ,λ, ). Covesely, let f S, (α,β,δ,λ, ). Usig Coollay 2.2, we have a ((λ δ)(β α)( )) (α 1). (1 α β) We may set ξ = ((λ δ)(β α)( )) (α 1) a (1 α β) ad lettig ξ = 1 ξ, we have 49

10 Aas Aljaah & Maslia Daus f ()= + a = = 1 ξ = ξ f ()+ + ξ ξ + ξ f (). + This comletes the oof of the theoem. Refeeces ξ + ξ (1 α β) ((λ δ)(β α)( )) (α 1) (1 α β) ((λ δ)(β α)( )) (α 1) Altitas O., Ima H. & Sivastava H. M Family of meomohically uivalet fuctios with ositive coefficiets. PaAme J. Math. 5: Ramada S. F. & Daus M O the Feete-Sego iequality fo a class of aalytic fuctios defied by usig geealied diffeetial oeato. Acta Uivesitatis Aulesis 26: Ualegaddi B. A. & Gaigi M. D Meomohic multivalet fuctios with ositive coefficiets. Neali Math. Sci. Re. 11(2): Cho N. E., Lee S. H. & Owa S A class of meomohic uivalet fuctios with ositive coefficiets. Kobe J. Math. 4: Cho N. E., Owa S., Lee S. H. & Altitas O Geealiatio class of cetai meomohic uivalet fuctios with ositive coefficiets. Kyugoo J. Math. 29: Jili L Poeties of some families of meomohic -valet fuctios. Mathematica Jaoicae 52(3): Liu J. L. & Sivastava H. M A liea oeato ad associated families of meo-mohically multivalet fuctios. J. Math. Aal. Al 259: Liu J. L. & Sivastava H. M Classes of meomohically multivalet fuctios associated with the geealied hyegeometic fuctio. Mathematical ad Comute Modellig 39(1): Joshi S. B., Kulai S. R. & Sivastava H. M Cetai classes of meomohic fuctios with ositive ad missig coefficiets. Joual of Mathematical Aalysis ad Alicatios 193(1): Raia R. K. & Sivastava H. M A ew class of meomohically multivalet fuctios with alicatios to geealied hyegeometic fuctios. Mathematical ad Comute Modellig 43(3): Aouf M. K. & Shammay A. E A cetai subclass of meomohically valet covex fuctios with egative coefficiets. J. Aox. Theoy ad Al. 1(2): Aouf M. K A family of meomohically -valet fuctios with ositive coefficiets. Geeal Mathematics 20(1): School of Mathematical Scieces Faculty of Sciece ad Techology Uivesiti Kebagsaa Malaysia UKM Bagi Selago DE, MALAYSIA aasjah@yahoo.com, maslia@um.edu.my* * Coesodig autho 50