The aim of this paper is to give a distortion estimation and the radius of starlikeness of the class S*K(A; B)

Aydogan, Melike

Polatoglu,  Yasar

Sen, Arzu

English

Aurel Vlaicu University of Arad Editing House: Journals

Theory and Applications of Mathematics & Computer Science 1 (2) (2011) 89–92Distortion Estimate and the Radius of Starlikeness of Janowski Close-to-StarFunctionsArzu Sena, Melike Aydoğanb,∗, Yaşar Polatoğluaa İstanbul Kültür University, Department of Mathematics and Computer Sciences İstanbul, Turkey.bIsik University, Sile Kampusu, Department of Mathematics, İstanbul, Turkey.AbstractLet F be the class of all analytic functions in the open unit disc D = {z| |z| < 1} of the form f (z) = z + a2z2 + a3z3 + · · · . Letg(z) be an element of F such that g(z) satisfies the condition(zg′(z)g(z)) =1 + Aφ(z)1 + Bφ(z),for all z ∈ D, where φ(z) is analytic in D and satisfying the conditions φ(0) = 0, |φ(z)| < 1, and −1 ≤ B < A ≤ 1, then g(z) is calledJanowski Starlike function in D. The class of such functions is denoted by S∗(A, B) (Janowski, 1973).The aim of this paper is to give a distortion estimation and the radius of starlikeness of the class S∗K(A, B).Keywords: Close-to-star function, the radius of starlikeness, distortion estimate.2000 MSC: 30C45.1. IntroductionLet Ω be the family of functions φ(z) which are regular in D and satisfy the condition φ(0) = 0, |φ(z)| < 1for all z ∈ D. The family of functions p(z) = 1 + p1z + p2z2 + · · · analytic in D, and satisfying the conditionsp(0) = 1, Re p(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. The class P is the Caratheodory class (Nehari, 1952).Next, for arbitrary fixed real numbers A and B which satisfy −1 ≤ B < A ≤ 1, we say p(z) belongs tothe class P(A, B) if p(z) = 1 + p2z2 + p3z3 + · · · is analytic in D and p(z) is given byp(z) =1 + Aφ(z)1 + Bφ(z), (1.2)∗Corresponding authorEmail addresses: a.sen@iku.edu.tr (Arzu Sen ), melikeaydogan43@gmail.com (Melike Aydoğan),y.polatoglu@iku.edu.tr (Yaşar Polatoğlu)90 Arzu Sen et al. / Theory and Applications of Mathematics & Computer Science 1 (2) (2011) 89–92for some φ(z) ∈ Ω and every z ∈ D. The class P(A, B) was introduced by (Janowski, 1973).Let f (z) be an element of F and g(z) be an element of S∗(A, B), if the conditionRe(f (z)g(z)) > 0 (1.3)is satisfied, then f (z) is called Janowski close-to-star function in D. The class of such functions is denotedby S∗K(A, B). The class of S∗K(A, B) is not empty because,f (z) =z(1 + Bz)A−BB . 1+z1−z ; B , 0,zeAz 1+z1−z ; B = 0.(1.4)Finally, let F(z) = z + α2z2 + α3z3 + · · · and G(z) = z + β2z2 + β3z3 + · · · be analytic functions in D, ifthere exist a function φ(z) ∈ Ω such that F(z) = G(φ(z)) for every z ∈ D, then we say that F(z) is subordinateto G(z), and we write F(z) ≺ G(z). We also note that if F(z) ≺ G(z), then F(D) ⊂ G(D).2. Main ResultsLemma 2.1. Let f (z) be an element of S∗K(A, B), thenRe[(1 + Bz)−A−BBf (z)z] > 0, B , 0, (2.1)Re[e−Azf (z)z] > 0, B = 0. (2.2)Proof. Since the function,g(z) =z(1 + Bz)A−BB ; B , 0,zeAz; B = 0.(2.3)belongs to the class S ∗(A, B) (Janowski, 1973), then using the definition of the class S∗K(A, B),Re(f (z)g(z)) = Re[f (z)z(1 + Bz)A−BB] = Re[(1 + Bz)−A−BB .f (z)z] > 0, B , 0, (2.4)Re(f (z)g(z)) = Re[f (z)zeAz] = Re[e−Azf (z)z] > 0, B = 0. (2.5)Theorem 2.2. Let f (z) be an element of S∗K(A, B), then for r = |z|F(−A,−B, r) ≤ | f (z)| ≤ F(A, B, r), B , 0, (2.6)G(−A, r) ≤ | f (z)| ≤ G(A, r), B = 0, (2.7)Arzu Sen et al. / Theory and Applications of Mathematics & Computer Science 1 (2) (2011) 89–92 91whereF(A, B, r) =r(1 + r)(1 + Br)A−BB(1 − r)(2.8)andG(A, r) =r(1 + r)eAr(1 − r). (2.9)Proof. Let g(z) be an element of S∗(A, B), thenF1(−A,−B, r) ≤ |g(z)| ≤ F1(A, B, r), (2.10)whereF1(A, B, r) =r(1 + Br)A−BB ; B , 0,reAr; B = 0.(2.11)On the other hand, if p(z) ∈ P then we have,1 − r1 + r≤ |p(z)| ≤1 + r1 − r(2.12)(Goodman, 1983).Considering 2.10 and 2.12 together and after the straightforward calculations, we get 2.6 and 2.7.We also note that the inequalities 2.6 and 2.7 are sharp because the extremal functions are;(1 + Bz)−A−BBf (z)z= p(z) =1 + z1 − z⇒ f (z) =z(1 + z)(1 + Bz)A−BB1 − z, B , 0, (2.13)e−Azf (z)z= p(z) =1 + z1 − z⇒ f (z) =z(1 + z)eAz1 − z, B = 0. (2.14)Remark. If we give the special values to A and B, we obtain that new inequalities and new growth theoremsfor the subclass of S ∗K(A, B). The special values of A and B can be ordered in the following manner:i. A = 1 , B = −1;ii. A = 1 − 2α, B = −1, 0 ≤ α < 1;iii. A = 1, B = 0;iv. A = α, B = 0, 0 < α < 1;v. A = 1, B = −1 + 1M , M >12 ;vi. A = α, B = −α, 0 < α < 1.Theorem 2.3. The radius of starlikeness of the class S∗K(A, B) is the smallest positive roots r0 of theequations, Q1(r) = −2r(1 − Br) + (1 − r2)(1 − Ar); B , 0,Q2(r) = −2r + (1 − Ar)(1 − r2); B = 0. (2.15)92 Arzu Sen et al. / Theory and Applications of Mathematics & Computer Science 1 (2) (2011) 89–92Proof. Using Lemma 2.1, then we obtain(1 + Bz)−A−BBf (z)z= p(z), B , 0, (2.16)e−Azf (z)z= p(z), B = 0, (2.17)z f′(z)f (z) =1+Az1+Bz + zp′(z)p(z) ; B , 0,z f′(z)f (z) = (1 + Az) + zp′(z)p(z) ; B = 0.(2.18)Thus,  Re(z f′(z)f (z) ) ≥ Min|z|=rRe(1+Az1+Bz ) + Min|z|=r,p(z)∈P Re(zp′(z)p(z) ); B , 0,Re(z f′(z)f (z) ) ≥ Min|z|=rRe(1 + Az) + Min|z|=r,p(z)∈P Re(zp′(z)p(z) ); B = 0.(2.19)Re(zf ′(z)f (z)) ≥ Min|z|=rRezp′(z)p(z)+ Min|z|=rRez(A − B)1 + Bz+ Min|z|=rRe(1). (2.20)Therefore we have: Re(zf ′(z)f (z) ) ≥−2r(1−Br)+(1−r2)(1−Ar)(1−r2)(1−Br) ; B , 0,Re(z f′(z)f (z) ) ≥−2r+(1−Ar)(1−r2)(1−r2) ; B = 0.(2.21)The denominator of the expression on the right hand sides of the inequalities 2.20 is positive for0 ≤ r < 1, Q1(0) = 1, Q(1) = −2(1 − B) < 0, Q2(0) = 1, Q2(1) = −2 < 0.Thus using intermediate value theorem and mean value theorem, the smallest positive roots of the equa-tions 2.15 lies between 0 and 1.Therefore the inequalityRe(zf ′(z)f (z))> 0, (2.22)is valid for r = |z| < r0. Hence the radius of starlikeness for S∗K(A, B) is not less than r0. The theorem isproved.ReferencesGoodman, A. W. (1983). Univalent functions, Volume I and II. Mariner publishing company, Tampa Florida, USA.Janowski, W. (1973). Some extremal problems for certain families of analytic functions. I. Annales Polonici Mathematici 28, 297–326.Nehari, Z. (1952). Conformal Mapping. McGraw-Hill, New York.

Distortion Estimate and the Radius of Starlikeness of Janowski Close-to-Star Functions

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Distortion Estimate and the Radius of Starlikeness of Janowski Close-to-Star Functions

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