Higher Order Close-to-Convex Functions related with Conic Domain

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1 Appl Math Inf Sci 8, No 5, Applied Mathematics & Infomation Sciences An Intenational Jounal Highe Ode Close-to-Convex Functions elated with Conic Domain Khalida Inayat Noo and Muhammad Aslam Noo Mathematics Depatment, COMSATS Institute of Infomation Technology, Pak oad, Islamabad, Pakistan eceived: 3 Sep 3, evised: Dec 3, Accepted: Dec 3 Published online: Sep Abstact: In this pape we define and study a class of analytic functions which map the open unit disk onto same conic egions and ae elated to Bailevic and highe ode close-to-convex functions We investigate ate of gowth of coefficients, Hankel deteminant poblem, inclusion esult and establish univalence citeion Some othe inteesting popeties of this class ae also studied Keywods: close-to-convex, univalent, conic egions, bounded bounday otation, Caatheoday functions, convolution, subodination AMS Subject Classification: 3C5, 3C5 Intoduction Let A be the class of functions analytic in the open unit disc E { : <} and be given by f+ n a n n Let S A be the class of functions which ae univalent and also K,S,C be the well known subclasses of S which, espectively, contain close-to-convex, stalike and convex functions Let V m ρ,m,ρ <, be the class of functions f analytic and locally univalent in E and satisfying the condition f ρ ρ dθ m f When ρ, we obtain the class V m m of functions with bounded bounday otation, see [] The class V m ρ was intoduced and discussed in some detail in [8] It can easily be shown that f V m ρ if and only if thee exists f V m such that f f ρ 3 The convolution of two functions f given by and g+ b n n is defined as n f gg f+ n a n b n n Coesponding autho khalidanoo@hotmailcom In [5], the domain Ω k,k [, is defined as follows: Ω k {u+iv : u>k u + v } Fo fixed k, Ω k epesents the conic egion bounded, successively, by the imaginay axis k, the ight banch of a hypebola <k< and a paabola k and an ellipsek> Also, we note that, fo no choice of kk>, Ω k educes to a disc, see [5,8] In this pape we will choose k [,] Fo k [,], the following functions, denoted by p k, ae univalent in E, continuous as egad to k, have eal coefficients and map E onto Ω k such that p k, p k >: p k +,k, + log +, k, + k sinh[ accosk actan ], <k< see[5] 5 Let P denote the class of Caatheodoy functions of positive eal pat Then the class Pp k P is defined as follows Definition Let p be analytic in E with p Then p Pp k, if p is subodinate to p k given by 5 We wite p P pk implies p p k in E, and pe p k E We note that Pp P It is easy to veify that Pp k is a convex set and Pp k Pρ, ρ k k+, c NSP

2 56 K I Noo, M A Noo: Highe Ode Close-to-Convex Functions elated whee Pρ is the class of functions with eal pat geate than ρ, see [6] Also, fo p Pp k, it is known [] that ag p < σ, k, actan k, k,, k We extend the class Pp k as given below Definition Let p be analytic in E with p Then p P m p k, if and only if, fo m, k [,], we have m p + m p p, 7 p, p Pp k When k, we obtain the class P m intoduced and studied in [] Also P p k Pp k We now define the following Definition 3 Let f A Then f k UV m, k [,], m if and only if [ + ] f f P m p k, E k UV m is called the class of functions with k-unifom bounday otation Fo k, UV m V m, see [,,3,] The coesponding class k U m is defined as We note that: k U m {F A : F f, f k UV m } i k UV k UCV, is the class of unifomly convex functions ii k U k ST is the class of unifomly stalike functions Fo details of these special case, we efe to [] Definition Let f A Then f k UT m if thee exists g k UV m such that f g Pp k, E Fo k, we have the class T m, intoduced and discussed in [] Also, fo k,m, k UT m educes to well known class K of close-to-convex functions, see [7] Let φ A Then f k UT m φ if and only if f φ k UT m fo E 6 Definition 5 Let f A Then, fo a, <, f k UT m a,,φ if and only if thee exists g k UT m φ such that We note that f +afa+g 8 k UT m,, k UT m, UT m,, T m Also UT,, log C, the class of quasi-convex functions, see [7] Thoughout this pape, we assume that k [,],,], m, a >, E, unless othewise specified Peliminaies Lemma [9] Let q, be analytic in E with q If α,c, θ < θ,e iθ, then implies θ θ { } q+ αq dθ > cα+ q θ θ qdθ > Lemma [6] Let f k U m Then thee exist s i k ST,i, such that f s m+ s m Lemma 3[] Let g V m ρ Then, fo ρ <,θ < θ, i g g ρ, g V m ii θ { } g g dθ > m ρ θ 3 Main esults Theoem Let G k UT m Then, fo θ < θ, e iθ, θ θ { G G } m dθ > k+ + σ c NSP

3 Appl Math Inf Sci 8, No 5, / wwwnatualspublishingcom/jounalsasp 57 Poof Since G k UT m, thee exist G k UV m and k UV m V m ρ, ρ k+ k, such that G G hσ, 9 whee σ is given by 6 and h P Also we obseve that, fo h P, θ agheiθ { θ θ } ilnhe iθ }, { e iθ h e iθ he iθ and so θ { e iθ h e iθ } he iθ dθ aghe iθ aghe iθ Hence max h P θ max h P { i e θ h e iθ } he iθ dθ aghe iθ aghe iθ θ sin cos Now diffeentiating 9 logaithmically and using Lemma 3 togethe with, we obtain θ θ { G G This completes the poof } m dθ > k+ + σ Theoem Let f k UT m a,,φ, a,<, θ < θ and e iθ Then θ θ whee { } { } p+ p m dθ > a+ p k+ + σ, p f f Poof Let Gg φ Then, by definition, f +afa+g, G k UT m Diffeentiating logaithmically, and with simple computations, we have a+ f f +a f f G G + That is, with p f f, we have {p+ p a+ p } G G Using Theoem, we obtain the equied esult [ ] Coollay Fo m σk+ +, we use Lemma to have fom Theoem, θ θ { f } dθ >, f k UT m a,,φ f Coollay Let f k UT m,,φ Then, fo m{ σk+ +}, θ θ { f f } dθ >, and hence f is univalent in E, see [7] If k, then σ and in this case f is univalent in E fo m Theoem 3 Fo < <, E, k UT a,,φ k UT a,,φ Poof Let f k UT a,,φ Then f +af a+g, Gg φ k UT, a+h, whee H G, H h φ We now show that H k UT and this will pove that f k UT a,,φ Now H G, G k UT, < Since G k UT, thee exists a function G g φ k UV such that G G Pp k in E Let G G, < It is easy to veify that G k UV in E Thus H G G G Ppk, since < This completes the poof c NSP

4 58 K I Noo, M A Noo: Highe Ode Close-to-Convex Functions elated emak Fom definition 5, the following integal epesentation fo the class k UT m a,,φ can easily be obtained A function f k UT m a,φ, if and only if thee exists a function G k UT m,,φ, such that f a+ a a a Gtdt Theoem Let f T m a,,φt m a,,φ Then f is a Bailevic function and hence univalent in < m, whee m is given by m { } m m Poof We can wite, fo f T m a,,φ, f a+ a t a Ftdt, F T m,,φ Let ac+id, c> Then we have f c++id c+id t c ptgtt id dt, 3 whee p P, g U m m We define g c+ G Then G G + g c+ c+ g Now g g P m and P m is a convex set, so G m and, it is known [] that G m is stalike fo < m whee m is given by Futhe we define f as f c++id G c+ tptt id dt c++id f is Bailevic function, see [], and hence univalent in < m Theefoe f, < m We note that f a+ f, ac+id This means that f, given by 3, is analytic and fo a+, it is possible to select unifom banch which f takes the value one fo and which is analytic fo < m and also allows us to compute the deivative in < m Thus we conclude that f is univalent in < m, whee m is given by This completes the poof Theoem 5 Let f UT m,,φt m,,φ Then the adius m of the disc which f maps onto a stalike domain is given by } {m m,, m, m, whee m m+ and Poof f T m,,φ implies that fg, Gg φ T m 5 Logaithmic diffeentiation of 5 gives us f f G G + Theefoe, using a esult [] fo G T m, we obtain { f } m++ f, and ight hand side is positive fo < m This poves the esult We now investigate the ate of gowth of coefficients of f k UT m a,,φ Let f be given by and let g + b n n, φ+ c n n n We have: n Theoem { 6 Let f} k UTa,,φ Then, fo m> σk+ σ, a n Cm,,k a+ n+a n{ k+ m++σ },n, whee Cm,,k is constant depending only on m, and k Poof We can wite f +afa+g, 6 whee Gg φ k UT m This implies thee exists G k UV m such that G G h σ 7 Then, with e iθ, we have n+a a n { n f +af } e inθ dθ n a+g t dθ, G k UT m 8 c NSP

5 Appl Math Inf Sci 8, No 5, / wwwnatualspublishingcom/jounalsasp 59 Now, since G k UT m, thee exists G k UV m, such that G G h σ, h P, and σ is given by 6 Using Lemma togethe with the known esult [] that k ST S ρ, ρ k+ k, we have t ρ m + G t ρ m, t,t S 9 { Define, fo m> σ σ ρ }, e iθ I Then, fom 9 I ρ ρ G dθ, G k UT m t ρ m + t ρ m h σ dθ ρ m t ρ m + h σ dθ, whee we have used well-known distotion esult fo the stalike function t We now apply Holde s inequality, use subodination fo stalike functions and a esult due to Pommeenke [] fo h P to have, fo σ ρm+> σ, I ρ ρ m t ρ m + Cm,,k σ h dθ σ ρ m ++ σ, whee Cm,,k is a constant depending only on m,,k That is ρ m ++ σ I O, Now, with n, we have fom 8, a n Cm,,k a+ n+a n m+ k+ +σ, n This completes the poof σ We note the following special cases i Let f UT m,, UTm, Then ρ, and a n On m f o m>, and, fo f UT m,, UTm,, we get a n On m,m> ii Let k and f T m,, Then, fo m a n On m ++ On m +,n When we take, then iii a n On m + Let f T m,, Then, fo m a n On m +, and with, we obtain, fo m a n On m, n This esult is poved in [] See also [5] iv Let { f k UT m,,log Then, fo m> σ ρ }, a n On m+ k+ +σ With k,, we have σ and so, fo m> a n On m, n Also, if we take k, Then σ and so a n On m, n Theoem 7 Let f k UT m,,φ Denote by L, f, the length of the image of the cicle unde f, by A, f, the aea of f < and M, fmax fe iθ Then L, fom, flog, whee O is a constant Poof Since f k UT m,,φ, we have f G, Gg φ k UT m This implies that f k UT m So thee exists G k UV m V m ρ such that G G p Pp k Pρ θ c NSP

6 6 K I Noo, M A Noo: Highe Ode Close-to-Convex Functions elated Now, with e iθ, L, f Now whee f dθ G dθ G p dθ, G V m ρ, p Pρ G p H+G p dξ dθ, f H dθdξ + H G G p G dθdξ I +I I f H dθdξ, H G G + f given by, d n m k we have k+ n d n n, I f dθ H dθ m k+ nn a n ξ n n n n n a n n m k+, and fo n, dξ α n ξ n dξ log + m n k+ nn a n log + But A, f n a n n is the aea of the image of n < by w f Theefoe I m k+ A, f log + Also, since A, fm, f, we have I m + M, f log k+ We now estimate I p Pρ,ρ k+ k, implies that we can wite So Theefoe Also and hence p ρ p ρ I ρ I ρ ρ +e it dµt, eit p ρ Integating by pats gives us dµt e it e it dµt G e it dµtdθdξ ξ e it dµt, G Hdθ dξ ξ {G e iagg }dθ dξ ξ I [m ρ+ρ] M, f dξ ξ Fom, and, we obtain the desied esult We study ac-length poblem with a diffeent technique as follows Theoem { 8 Let f} k UT m,,φ Then, fo m> σk+, k+ m ++σ L, fo,, whee σ is given by 6 and O is a constant c NSP

7 Appl Math Inf Sci 8, No 5, / wwwnatualspublishingcom/jounalsasp 6 Poof We can wite f G, Gg φ k UT m G h σ, Gg φ k UT m,h P s m + m hσ, s,s k ST, 3 s by using Lemma Also s i k ST implies that s i S ρ,ρ k+ k,i, Theefoe, fo e iθ L, f f s k+ m + dθ s k+ m h σ dθ Since s is stalike and hence univalent, so we have L, f k+ m s k+ m + h σ dθ Holde s inequality togethe with subodination fo stalike functions, we have L, f k+ m O e iθ h dθ k+ m + k+ m ++σ, σ σ dθ σ fo m+ k+ > σ This completes the poof As special cases, we note the following i Fo,m and k, which gives us σ This gives us L, fo ii We take k and Then σ and f T m This gives us m + L, fo, We shall estimate the gowth ate of H q n fo the functions in the class UT m,,φ This is the main motivation of next esult Let f A and be given by Suppose that the qth Hankel deteminant of f is defined fo q, n by a n a n+ a n+q H q n a n+ a n+ a n+q a n+q Theoem 9 Let f UT m,,φ and let the qth Hankel deteminant of { f fo q,n, be defined by The, fo m 8q }, H q non m + q q,n, whee O is a constant depending upon,m and q only To pove Theoem 9, we need the following esults and fo these we efe to [9] Lemma Let f A and be given by and let the qth Hankel deteminant of f be defined by Then, witing j n j n,, f We have H q n q n q 3 n+ q n+q q 3 n+ q n+ q n+q, 5 q n+q q n+q whee, with n,, fa n, we define fo j, j n,, f j n,, f j n+,, f 6 Lemma 5 With x n n+ y,v and intege j n+v,v,x, f j j y l v l n l n+ l j l n+v,v+l,y, f l PoofTheoem 9 Since f UT m,,φ, we can wite f G, g UT m,gg φ 7 Now, fo G UT m, thee exists G UV m V m such that G Pp Also, fo p Pp, we have ag p < G which gives us σ Thus we can wite 7 as f [G p ] G p,g V m, p P s m + m p, s c NSP

8 6 K I Noo, M A Noo: Highe Ode Close-to-Convex Functions elated whee s,s S, whee we have used a esult due to Bannan [] Also we can choose a with such that fo any univalent functions s max s, 8 see [3] Now, fo j, any noneo complex numbe, conside j n,, f n+ j j s s m + m Thus, fo m+ 8 j+, j n,, f n+ j n+ j p dθ j s m + s m p dθ j m s m + j p dθ j m n+ j p dθ s m + j j +3 Cm,, j { } [ m + j] m j+ O, O is a constant and we have used 8, distotion esults fo stalike functions, Holde s inequality and a esult fo h P, see [] Choosing n, we have, fo m+ 8 j+, j n,, f On m + j, and using Lemma 5, we obtain j n,e iθ n, fon m + j, n 9 We use Lemma and follow the simila agument given in [9], to have H q non m + q q, n fo m+ 8q This completes the poof Special Case When, m 6, we have a n On m and Fo this case we note that Acknowledgement H q non m q q, n H non m, m The authos ae gateful to D S M Junaid Zaidi, ecto, COMSATS Institute of Infomation Technology, Pakistan fo poviding excellent eseach and academic envionment efeences [] I E Bailevic, On a class of integability in quadatues of the Loewne-Kufaev equation, Math Sb, 37, [] D A Bannan, On functions of bounded bounday otations, Poc Edin Math Soc,, [3] G Golusin, On distotion theoems and coefficients of univalent functions, Mat Sb, 9, [] A W Goodman, Univalent Functions, Vol I,II, Polygonal Publishing House, Washington, New Jesey, 983 [5] S Kanas, Techniques of diffeential subodination fo domains bounded by conic sections, Intenat J Math Math Sci, 38, [6] S Kanas, A Wisniowska, Conic egions and k-unifom convexity, J Comput Appl Math, 5, [7] W Kaplan, Close-to-convex Schlicht functions, Michigan Math J,, [8] E J Moulis, Genealiations of the obetson functions, Pacific J Math, 8, [9] J W Noonan, D K Thomas, On Hankel deteminant of aeally mean p-valent functions, Poc Lond Math Soc, 5, [] K I Noo, On subclasses of close-to-convex functions of highe ode, Intenat J Math Math Sci, 6, [] K I Noo, On genealiations of close-to-convexity, Mathematica, 3, [] K I Noo, Some ecdent developments in the theoy of functions with bounded bounday otation, Inst Math Comput Sci,, [3] K I Noo, W Ul-Haq, M Aif, S Mustafa, On bounded bounday and bounded adius otation, J Inequal Appl, 9, Aticle ID 83687, pages 9 [] K I Noo, Some popeties of analytic functions with bounded adius otation, Complex Va Elliptic, Equat, 5, [5] K I Noo, Highe ode close-to-convex functions, Math Japonica, 37, c NSP

Appl Math Inf Sci 8, No 5, 55-63 / wwwnatualspublishingcom/jounalsasp 63 [6] K I Noo, On a genealiation of unifomaly convex and elated functions, Comput Math Appl, 6, 7-5 [7] K I Noo, On quasi-convex

Pavatham, S adha, On cetain classes of analytic functions, Ann Polon Math, 9, 3-3 98 [] B Pinchuk, Functions of bounded bounday otation, Is J Math,, 7-6 97 [] Ch Pommeenke, On stalike and

9 Appl Math Inf Sci 8, No 5, / wwwnatualspublishingcom/jounalsasp 63 [6] K I Noo, On a genealiation of unifomaly convex and elated functions, Comput Math Appl, 6, 7-5 [7] K I Noo, On quasi-convex univalent functions and elated topics, Intenat J Math Sci,, [8] K I Noo, Fayya, M A Noo Some classes of k- unifomly functions with bounded adius otation, Appl Math Inf Sci, 8, No, [9] Pavatham, S adha, On cetain classes of analytic functions, Ann Polon Math, 9, [] B Pinchuk, Functions of bounded bounday otation, Is J Math,, [] Ch Pommeenke, On stalike and close-to-convex functions, Poc London Math Soc, 3, [] F onning, Unifom convex functions and a coesponding class of stalike functions, Poc Ame Math Soc, 8, Khalida Inayat Noo is a leading wold-known figue in mathematics She obtained he PhD fom Univesity of Wales SwanseaUK She has a vast expeience of teaching and eseach at univesity levels in vaious counties including Ian, Pakistan, Saudi Aabia, Canada and United Aab Emiates She was awaded HEC best eseach pape awad in 9 and CIIT Medal fo innovation in 9 She has been awaded by the Pesident of Pakistan: Pesident s Awad fo pide of pefomance on August, fo he outstanding contibutions in mathematical sciences and othe fields He field of inteest and specialiation is Complex analysis, Geometic function theoy, Functional and Convex analysis She intoduced a new technique, now called as Noo Integal Opeato which poved to be an innovation in the field of geometic function theoy and has bought new dimensions in the ealm of eseach in this aea She has been pesonally instumental in establishing PhD/MS pogams at COMSATS Institute of Infomation Technology, Pakistan D Khalida Inayat Noo has supevised successfully seveal PhD and MS/MPhil students She has been an invited speake of numbe of confeences and has published moe than Fou hunded eseach aticles in eputed intenational jounals of mathematical and engineeing sciences She is membe of editoial boads of seveal intenational jounals of mathematical and engineeing sciences M Aslam Noo eaned his PhD degee fom Bunel Univesity, London, UK 975 in the field of Applied Mathematics Numeical Analysis and Optimiation He has vast expeience of teaching and eseach at univesity levels in vaious counties including Pakistan, Ian, Canada, Saudi Aabia and United Aab Emiates His field of inteest and specialiation is vesatile in natue It coves many aeas of Mathematical and Engineeing sciences such as Vaiational Inequalities, Opeations eseach and Numeical Analysis He has been awaded by the Pesident of Pakistan: Pesident s Awad fo pide of pefomance on August, 8, in ecognition of his contibutions in the field of Mathematical Sciences He was awaded HEC Best eseach pape awad in 9 He has supevised successfully seveal PhD and MS/MPhil students He is cuently membe of the Editoial Boad of seveal eputed intenational jounals of Mathematics and Engineeing sciences He has moe than 75 eseach papes to his cedit which wee published in leading wold class jounals c NSP

KOEBE DOMAINS FOR THE CLASSES OF FUNCTIONS WITH RANGES INCLUDED IN GIVEN SETS

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