Aydogan, Melike

Polatoglu, Yasar

Yemisci, Arzu

Bulgarian Digital Mathematics Library at IMI-BAS

M a t h e m a t i c aB a l k a n i c a—————————NewSeries Vol. 26, 2012, Fasc. 1-2Growth Theorem and the Radius of Starlikenessof Close-to-Spirallike FunctionsYas¸ar Polatog˜lu 1, Melike Aydog˜an 2, Arzu Yemisci 3Presented at 6th International Conference “TMSF’ 2011”Let A be the class of all analytic functions in the open unit disc D = {z| |z| < 1} ofthe form f(z) = z + a2z2 + a3z3 + · · · . Let g(z) be an element of A and satisfy the conditionRe(eiα g′(z)g(z)) > 0 for some α, |α| < pi2. Then g(z) is said to be α-spirallike. Such functions areknown to be univalent in D. It was shown by L. Spacek [11], that the α-spirallike functionsare univalent in |z| < 1.Let S∗α denote the class of all functions g(z) satisfying the above condition for a givenα. A function f(z) ∈ A is called close-to-α spirallike, if there exists a function g(z) in S∗α suchthat Re( f(z)g(z)) > 0. The class of such functions is denoted by S∗αK.The aim of this paper is to give a growth theorem and the radius of starlikeness of theclass S∗αK.AMS Subj. Classification: 30C45Key Words: close-to-α spirallike functions, growth theorem, radius of starlikeness1. IntroductionLet Ω be the family of functions φ(z) which are regular in D and satisfyingthe condition φ(0) = 0, |φ(z)| < 1 for all z ∈ D. The family of functionsp(z) = 1+p1z+p2z2+ · · · analytic in D, and satisfying the conditions p(0) = 1,Rep(z) > 0 is denoted by P such that p(z) in P if and only ifp(z) =1 + φ(z)1− φ(z) (1.1)for some φ(z) ∈ Ω, and every z ∈ D. Then we say that p(z) ∈ P is theCaratheodory function, see [1].212 Y. Polatog˜lu, M. Aydog˜an, A. YemisciNext, let A be the family of functions which are analytic in the open unitdisc D = {z : |z| < 1}. Let g(z) be an element of A and satisfying the condition,Re(eiαg′(z)g(z)) > 0 (1.2)for some α, |α| < 12 , then g(z) is said to be α-spirallike. The class of suchfunctions is denoted by S∗α, see [2]. A function f(z) ∈ A is close-to-α-spirallikeif there exists a function g(z) in S∗α such thatRe(f(z)g(z)) > 0 (1.3)for all z ∈ D. The class of such functions is denoted by S∗αK. By takingα = 0, we see that every close-to-α-spirallike functions reduces to a close-to-star function, see [6]. Close-to-star functions are not always univalent in D;consequently, close-to-α-spirallike functions need not to be univalent in D. Wealso note that if g(z) ∈ S∗α, the introduction of appropriate normalizing factorsenables us to writesecα[eiαzg′(z)g(z)− i sinα]z=0 = 1. (1.4)This leads to a useful representation formula for being member of S∗α in termsof functions P. The function g(z) is α-spirallike function if and only if thereexists a function p(z) in P ,eiαzg′(z)g(z)= cosαp(z) + i sinα. (1.5)Many geometric and analytic properties of α-spirallike functions can be obtainedfrom (1.5). Using (1.5), D. Pashkouleva [8] found the radius of spiral convexityof a close-to-spirallike functions.Finally, we need to give a description. Let F (z) = z + α2z2 + α3z3 + · · ·and G(z) = z + β2z2 + β3z3 + · · · be analytic functions in D, if there exists afunction φ(z) ∈ Ω such that F (z) = G(φ(z)) for every z ∈ D, then we say thatF (z) is subordinate to G(z), and we write F (z) ≺ G(z). We also note that ifF (z) ≺ G(z), then F (D) ⊂ G(D). The following lemma is due to I. S. Jack, see[2] and plays very important role in our proof of Theorem 2.2.Lemma 1.1. Let w(z) be regular in the unit disc with w(0) = 0. Thenif |w(z)| obtains its maximum value on the circle |z| = r at the point z1, onehas. z1w′(z1) = kw(z1), for some k ≥ 1.Growth Theorem and the Radius of Starlikeness . . . 2132. Main resultsLemma 2.1. ([7]) Let g(z) be an element of S∗α, thenr(1− r)cos2 α−cosα(1 + r)cos2 α+cosα ≤|g(z)| ≤ r(1− r)cos2 α+cosα(1 + r)cos2 α−cosα . (2.1)This inequality is sharp because the extremal function isf(z) = z(1− z)2 cosαe−iαwithζ =r(r − eiα)1− reiα ,thenζg′(z)g(z)=1− 2 cosα.r + e−2iαr21− r2 . (2.2)P r o o f. Let g(z) ∈ S∗α. If there exists a function p(z) in P such thateiαzg′(z)g(z)= cosαp(z) + i sinα. (2.3)By using p(z) is subordinate to (1+z1−z ), we obtain ([1]).∣∣∣∣p(z)− 1 + r21− r2∣∣∣∣ ≤ 2r1− r2 . (2.4)Equations (2.3) and (2.4) yield∣∣∣∣z g′(z)g(z) − 1 + e−2iαr21− r2∣∣∣∣ ≤ 2r cosα1− r2 . (2.5)which gives:1− (2 cosα)r + (cos 2α)r21− r2 ≤ Re(zg′(z)g(z)) ≤ 1 + (2 cosα)r + (cos 2α)r21− r2 . (2.6)On the other hand we haveRe(zg′(z)g(z)) = r∂∂rlog |g(z)| . (2.7)214 Y. Polatog˜lu, M. Aydog˜an, A. YemisciUsing (2.7) and after the simple calculations we get,1− (2 cosα)r + (cos 2α)r2r(1− r)(1 + r) ≤∂∂rlog |g(z)| ≤ 1 + (2 cosα)r + (cos 2α)r2r(1− r)(1 + r) . (2.8)then after integration we obtain (2.1).Theorem 2.2.g(z) ∈ S∗α ⇔ (zg′(z)g(z)− 1) ≺ 2(e−iα cosα)z1− z = F (z). (2.9)P r o o f. Let g(z) be an element of S∗α, then we define the function φ(z)byg(z)z= (1− φ(z))−2 cosαe−iα , (2.10)where (1 − φ(z))−2 cosαe−iα has the value 1 at z = 0, then φ(z) is analytic andφ(0) = 0. If we take logarithmic derivative from (2.10), we get(zg′(z)g(z)− 1) = (2 cosαe−iα)zφ′(z)1− φ(z) . (2.11)Now it is easy to realize that the subordination is equivalent |φ(z)| < 1 for allz ∈ D. Indeed, assume to the contrary: Then there exists z1 ∈ D such that|φ(z1)| = 1. So from I. S. Jack Lemma, z1φ′(z1) = kφ(z1) for some k ≥ 1 andfor such z1 ∈ D, we have(z1g′(z1)g(z1)− 1) = (2 cosαe−iα)kφ(z1)1− φ(z1) = F (φ(z1)) /∈ F (D).But this contradicts (2.11); so our assumption is wrong, i.e., |φ(z)| < 1 for allz ∈ D. This shows that(zg′(z)g(z)− 1) ≺ (2e−iα cosα)z1− z .Conversely,(zg′(z)g(z)− 1) ≺ e−iα(2 cosα)z1− z ⇒zg′(z)g(z)− 1 = e−iα (2 cosα)φ(z)1− φ(z) ⇒Growth Theorem and the Radius of Starlikeness . . . 215eiαzg′(z)g(z)= cosα1 + φ(z)1− φ(z) + i sinα.This shows that g(z) ∈ S∗α.Corollary 2.3. Let g(z) be an element of S∗α, then∣∣∣∣( zg(z)) eiα2 cosα − 1∣∣∣∣ < 1. (2.12)P r o o f. Sinceg(z)z= (1− φ(z))−2 cosαe−iα . (2.13)is analytic and (1− φ(z))−2 cosαe−iα has the value 1 at z = 0, then after simplecalculations from (2.12) with |φ(z)| < 1 we get (2.13). We also note that theinequality (2.12) is the Marx-Strohhacker inequality for α-spirallike functions.If we take α = 0 we obtain, ∣∣∣∣( zg(z)) 12 − 1∣∣∣∣ < 1. (2.14)The inequality (2.14) is the Marx-Strohhacker inequality for starlike functions,see [5].Theorem 2.4. Let f(z) be an element of S∗αK, thenr(1 + r)cos2 α+cosα−1(1− r)cos2 α−cosα+1 ≤ |f(z)|≤ r(1 + r)cos2 α−cosα+1(1− r)cos2 α+cosα−1.This inequality is sharp because the extremal function isf(z) = z(1 + z)(1− z)−2 cosαe−iα−1withζ =r(r − eiα)1− reiα .P r o o f. Using the definition of close-to-α-spirallike function, we canwrite;Re(f(z)g(z)) > 0⇒ f(z)g(z)= p(z)⇒∣∣∣∣f(z)g(z)∣∣∣∣ = |p(z)| . (2.15)216 Y. Polatog˜lu, M. Aydog˜an, A. YemisciOn the other hand, since p(z) ∈ P, then we have;1− r1 + r≤ |p(z)| ≤ 1 + r1− r . (2.16)Considering the inequalities (2.15) and (2.16) together we get the result.Lemma 2.5. Let f(z) be an element of S∗αK, thenRe[(1− z)2 cosαe−iα f(z)z]> 0. (2.17)P r o o f. Sinceg(z) =z1− z)2 cosαe−iαis α-spirallike function, then using the definition of close-to-α-spirallike functionwe get (2.17).Theorem 2.6. ([7]) The radius of starlikeness of the class of S∗αK isr =1(1 + cosα) +√(1 + cosα)2 − cos 2α . (2.18)This radius is sharp because the extremal function isf(z) =z(1 + z)(1− z)2 cosαe−iα−1withζ =r(r − eiα)1− reiα .P r o o f. Using Lemma 2.5 and after straightforward calculations, we getzf ′(z)f(z)= zp′(z)p(z)+1 + (2 cosαe−iα − 1)z1− z . (2.19)On the other hand if we takew =1 + (2 cosαe−iα − 1)z1− z ,then we obtainRew ≥ 1− 2r cosαr + cos 2αr21− r2 . (2.20)Growth Theorem and the Radius of Starlikeness . . . 217Therefore, we haveRe(zf ′(z)f(z)) = Rezp′(z)p(z)+Rew ≥ −2r1− r2 +1− 2 cosαr + cos 2αr21− r2 ⇒Re(zf ′(z)f(z)) ≥ 1− 2(1 + cosα)r + cos 2αr21− r2 . (2.21)From (2.21) we conclude that the radius of starlikeness of S∗αK is the smallestpositive root of1− 2(1 + cosα)r + cos 2αr2,which is the result.Corollary 2.7. If we take α = 0, then we obtainr =12 +√3=2−√34− 3 = 2−√3. (2.22)This is the radius of starlikeness of the class of close-to-star functions whichwas obtained by Sakaguchi, see [10].References[1] A.W. Goodman, Univalent Functions, Volume I, Mariner Publishing, TampaFlorida, 1984.[2] I.S. Jack, Functions starlike and convex of order α. J. London. Math. Soc.2, No 3(1971), 469-474.[3] R.J. Libera, Univalent α-spiral functions. Canad. J. Math., 19 (1967), 449-456.[4] R.J. Libera, M.R. Ziegler, Regular functions f(z) for which zf ′(z) is α-spiral.Trans. Amer. Math. Soc., 166 (1972), 361-370.[5] O.R. Maxwell, On close-to-convex functions. Michigan Mathematical Jour-nal, 3, Issue 1 (1955/1956), 59-62.[6] S.S. Miller, P.T. Mocanu and M.O. Reade, On generalized Convexity inConformal mapping, II. Rev. Roum. Math. Pures. et Appl., Tome XXI,No 2 (1976), 219-225.[7] Z. Nehari, Conformal Mapping. Dover Publications, New-York, 1952.[8] D.Z. Pashkouleva, The radius of spiral-convexity of a class of spiral-likefunctions. C. R. Acad. Bulg. Sci., 30 (1977), 1675-1677.[9] M.S. Robertson, A characterization of the class of the class of starlike uni-valent functions. Michigan Math. J., 26, Issue 1 (1979), 65-69.218 Y. Polatog˜lu, M. Aydog˜an, A. Yemisci[10] K. Sakaguchi, A variational method for functions with positive real part. J.Math. Soc. Japan, 16 (1964), 287-297.[11] L. Spacˇek, Contribution a` la theorie des fonctions univalents. Gˇasopis Peˇst.Mat. Fys., 62 (1933), 12-19.1 Department of Mathematics and Computer SciencesI˙stanbul Ku¨ltu¨r University, I˙stanbul, TURKEYe-mail: y.polatoglu@iku.edu.tr2 Department of Computer EngineeringYeni Yuzyil UniversityCevizlibag, Topkapi, I˙stanbul, TURKEYe-mail: melike.aydogan@yeniyuzyil.edu.tr Received: June 28, 20113 Department of Mathematics and Computer SciencesI˙stanbul Ku¨ltu¨r University, I˙stanbul, TURKEYe-mail: a.sen@iku.edu.tr

Growth Theorem and the Radius of Starlikeness of Close-to-Spirallike Functions

http://sci-gems.math.bas.bg/jspui/bitstream/10525/2667/1/MB-26-211-218.pdf

Growth Theorem and the Radius of Starlikeness of Close-to-Spirallike Functions

Abstract

Similar works

Full text

Available Versions

Bulgarian Digital Mathematics Library at IMI-BAS