CtA, cc( ) ct,-l where S*() and C(-) denote the classes of starlike and convex 1-A

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1 Internat. J. Math. & Math. Sci. VOL. 14 NO. 4 (1991) ON RADII OF CONVEXITY AND STARLIKENESS OF SOME CLASSES OF ANALYTIC FUNCTIONS KHALIDA INAYAT NOOR Mathematics Department College of Science King Saud University Riyadh Saudi Arabia (Received December 5, 1989 and in revised form March 21, 1990) ABSTRACT. Let P[A,B],-1 :B <A 1, be the class of functionsp such that p(z) is subordinate to /_ Let P(at) be the class of functions with positive real part greater than a, 0 al < 1. It is clear that +Bz P[A,B]CP(_B CP[1,-1]. The principal results in this paper are the determination of the radius of -starlikeness and -convexity of f(z) with -_--, when f(z) is restricted to certain classes of univalent and analytic functions related vith P[A,B ]. KEY WORDS AND PHRASES. Subordinate, starlike and convex functions, bounded boundary rotation, radius, close-to-convex functions AMS SUBJECT CLASSIFICATION CODE. 30A32, 30A34, 30C INTRODUCTION. Let f be analytic in E {z "] z[ < 1}, and be given by f(z)-z +, a,,z". (1.1) A function g, analytic in E, is called subordinate to a function G if there exi.sts w(z) analytic in E with w(0) 0 and w(z)l < 1 in E, such that g(z) G(w(z)). a Schwarz function w(z), In [1], Janowski introduced the class P[A,b]. For A and B,-1 <B <A 1, a function p, analytic in I+A E with p(0) 1 belongs to the class P[A,B ifp(z) is subordinate to --;-,. Also C[A,B and S*[A,B denote the classes of functions, analytic in E and given by (1.1) such that (# (z)y z/"(z) a,-.-s-.p[a,b] and-.p[a,b] respectively. For A- 1, and B--1, we note that C[1,-1]= C and S*[1,-1] S*, the classes of convex and starlike functions in e. Also S*[A,B C. S*(-a) (2 S*[1,-1] and CtA, cc( ) ct,-l where S*() and C(-) denote the classes of starlike and convex functions of order respectively. These classes were first introduced by Robertson in [2]. A function f, analytic in E and given by (1.1), is said to be in the class R,[A,B ], -1 B <A 1, if and only if

2 K.I. NOOR Hence (zf (z)).zp (z) f Cz) 1-e p(z) 1-B Using Lemma 2.3 for a I, we have for RI Re[ (Zf (z)) 1-(3A -B)r +A2r f (z) 1-B :, (r)(1-br) l-a] A-B 1-(2 +A -B)r +Ar -B (1 -Ar)(1 -Br) t4 t)r -t and this implies that Re[ -,t) i-8 0 for Iz] < r0, where ro is given by (3.1). The inequality RI <R_, is satisfied whenever T(r)- 1-(2 +A -B)r +Ar2>O. But T(0)- >0 and T(1)-B <0. So T(r) has at least one root in (0,1). Let ro, given by (3.1) be that root of T(r) O. Then in [0,ro),Rl <Rz and hence fec( i ) for all z with zl r ro < 1. This result is sharp for the function f0 e S*[A,B zfo (Z) o(z) such that +Az + Bz THEOREM 3.Z. Let g e. S*[A,B z/ (z) and let e P[A,B ]. -A Then f e C(_---) for zl < ro, where ro is given by (3.1). PROOF. zf (z) g(z)p(z), pe P[A,B ]. This gives us Applying the usual inequalities, we obtain Re[(Zf (z)) (z/ (z)) / (z) x-a] f (z) 1-B zg (z) zp (z) + g(z) (z) -Ar (A -B)r 1 -A 1-Br (r)(1-br) 1-B (,4 -B)[1-(2 +a -B)r +Ar] (1 -B)(1 -Ar) (1 -Br) Hence we obtain the required result that f e C(_-) for Izl < ro and ro is given by (3.1). THEOREM 3.3. Let g e S*[A,B and P[A,B ]. Then e ro, where ro is given by (3.1). PROOF. We have zf (z) g(z)p(z), p P[A,B and so Thus Re[ (zf (z)) g (z) 1-B (zf (z)) g(z).p(z) + g (z) zg (z) Re p(z) (1 -Ar) (1 -Br) (A -B) (1 -B) zp (z) (1 -Br) (A-B)r (r) 1-(3A-B)r +A2r (1 -Ar) (1 -Br) 1 -(2 +A -B)r +Ar (r)(1-br)]-i-b

3 RADII OF CONVEXITY AND STARLIKENESS 743 (S,(z)) f(z)-, $,,SeS*[A,B]. (S(z))" (1.2) Clearly k a 2 and R2[A,B S*[A,B ]. Also Rk[ 1,-1] U,, the class of functions with bounded radius rotation discussed in [3]. Similarly we can define the class V,[A,B as follows. A function f, analytic in E and given by (1.1) belongs to V[A,B ], k :,. 2, if and only if From (1.2) and (1.3), it is clear that [/_ (S,Cz )lz y f (z)-, S,,S2eS*[A,B (1.3) (S(z )/z )" f e Vk[A,B if and only if zf e Rk[A,B (1.4) It may be noted that V2[A,B C[A,B and V,[1,-1] V,, the class of functions of bounded rotation first discussed by Paatero [4]. 2. PRELIMINARY RESULTS LEMMA 2.1 [5] Let p e P[A,B]. Then r 1 +Ar -B Re P(Z)lP(z)l +Br The following is the extension of Libera s result [6]. LEMMA 2.2. Let N and D be analytic in E, D map onto a many-sheeted starlike region. V (z N(z N(0) 0 D(0) and e P[A,B ]. Then D-z) e P[A,B ]. For the proof of this result we refer to [5]. where and LEMMA 2.3. [7] Let p e P[A,B ]. Then, for z e E, a 0 and a 0, we have This result is sharp. 3. MAIN RESULTS. zp (z) Re{ ap(z) + fi p(z) } KI ct-{a-b) + 2aA}r + ctaur (I -Ar)(1 -Br) 15 A +B 2[(L1KI)v2 13(1 -ABR2)] A -B + (A -B)(1 -r 2) R: r i -Br LI 1 -A)(1 +/r) ct(a -B )(1- ru) + I3(1-B )(1 +Br2). THEOREM 3.1. Let f S*[A,B ]. Then f e C(_--) for R -:R [Z[ <r (2 +A -B) +V (2 +A -B)2-4A This result is sharp. PROOF. We have zf (z) f(z)p(z), p e P[A,B (3.1)

4 744 K.I. NOOR.r p() fo Izl Hence e < ro, where r is given by (3.1). Our next result is about the radius of convexity problem for the class Vk[A,B ]. THEOREM 3.4. Let f e V,[A,B ], k 2. Then f e C(_--) for zl < rl, where PROOF. Since f e V,[A,B ], we have from (1.3) This implies that SO r + - k(1-b) + VCk2(1 -B) + 16/3 (s#)/z) / f (z).,_, S,S S*[A,n (S# )/z )" (zf (z)) f (z) (k 1)(k 1) Pt(Z)- -- Pc(z)" pl,p2 e P[A,B] Re[ (zf (z)) if (z 1-B "+ 1-Br "- +B" --B Hence f e C(_--) for Izl < r,, r, is given by (3.2). (A -B)-(1 -B )(A -B )r -B(A -B )r (1 -B)(1 -Br) (3.2) by From Theorem 3.4 and relation (1.4) we have the following: -,/l-a\ THEOREM 3.5. Let f e R[A,B ]. Then f e 5 () for zl < rt where rt is given by (3.2). THEOREM 3.6. Let a and m be any positive integers and f e Rk[A,B ]. Then the function F defined (F(z))" belongs tos*(i-) for Izl < r, r is given by (3.2). az,, + m. i t"-l(f(t))dt (3.3) PROOF. LetJ(z) / t"-t(f(t))"dt and so ((z))". a m. j(z) and or zf (z) F(z) o2f (z) F(z) zj (z) J(z) 1 zj (z)-mj(z) N(z) ct J(z) D(z) N(0)-- 0-D(0)

5 RADII OF CONVEXITY AND STARLIKENESS 745 By a result of Bernardi [8] and Theorem 3.5,D(z) is a (m + ct- 1)-valent starlike function for [z[ < rl. Also N (z)- 1 D (z) ct { (zj (z)) -mj (z)} r (z) z[ (z) -A -A Now, by Theorem 3.5, f e S*( _--) for zl < r, d thi implies that,v.t,, e p(r-z) for zl < rt. Hence D(z)N(z) P()I e for -B [zl<r,, see t8]. This proves our result. Similarly, we can prove the following: THEOREM 3.7. Let a and m be positive integers and f V[A,B ]. Let Fbe defined by (3.3). Then f e C(-- s) for Izl < rl where r is given by (3.2). We now prove: THEOREM 3.8. Let fand g R[A,B and, for m positive integers, let F be defined as Then F S*(y)forlzl <to (F(z)) (m + ct) (g(z))" i t" l(f(t))adt (3.4) where r0 Then So min(r, r2), r is given by (3.2) and r is the least positive root of the equation {(1-B)-a()-{(A-B(1 + 2m)}r +{(A-B)+2m(A-B)+cg, l-a)}r2-o, PROOF. LetJl(z)- /--" / t -l(f(t))dt. zm (F(z))"-(f-G) Jt(z), where by Theorem 3.6,Jr es*(_-_-)forlzl <r. (3.5) Thus F(z"-- J(z - + m g(z) Re F(z) 1-B > l + l,,_b r /(l+r) + --(B-A)r /(1-R) "{ {(I-B-a+aA)+[(B-A)(I+2m)]r+[(A-B)+2m(A-B)+a()]r} -B)(1 -r) zf (z) -A i impi, or Izl Re- < r, where r i th,,at poitiv, root o 3.5). H,n,::, }a(x [z[ < r0, where r min(r 1, r2). Similarly, we have the following: TItEOREM 3.9. Let f and g e V[A,B and, for it, m positive integers, let F be defined by (3.4). Then F e C(_---) for [z[ <ro, where ro is as given in Theorem 3.8. THEOREM Let g e V[A,B and" e P[A,B] and let F be defined by

6 746 K.I. NOOR where m is any positive integer. F(z) re+l/ t -lf(t)dt, Then there exists a function G such that for zl < r,, where rl is given by (3.2). PROOF. Let G (z) e P 1-B G(z) G e C 1-B m+l [ t -lg(t)dt. Then, by Theorem 3.7 with ct 1 G e C( Z-A _--) for zl < rt and rt is defined by (3.2). Now z f(z)-m([t -lf(t)dt z q(z)-m( [ t - g(t)dt) t f (t)dt N(z) O(z) Also Thus, by Lemma 2.2, we have -i7 N (z) D (z) f (z) ep[a,b] for g (z) zl <rl tq, ep[a,b] C P(-s) for Izl < r, and this proves our result. REFERENCES 1. JANOWSKI, W. Some extremel problems for certain families of analytic functions, Ann. Polon. Math. 28 (1973), ROBERTSON, M. S. On the theory of univalent functions, Ann. Math. 37 (1936), KARUNAKARAN, V. and PADMA, K. Functions of bounded radius rotation, Indian J. Pure Appl. Math, 12 (1981), PAATERO, V. Uber die Konforme Abbildung von Gebieten deren Rander yon beschrankter Drehung sind, Ann. Acad. Sci. Fenn. Ser. A33 (1933), PARVATHAM, R. and SHANMUGAM, T. N. On analytic functions with reference to an integral operator, Bull, Austral, Math. Soc. 28 (1983), LIBERA, R. J Some classes of regular univalent functions, Proc. Amer. Math. Soc. 16 (1965), 7. ANH, V. V. and TUAN, P. D. Extremal problems for a class of functions of positive real part and applications, J. Aust. Math. Soc, 41 (1986), BERNARDI, S. M. Convex and starlike univalent functions, Trans. Amer, Math. Sot._ 135 (1969),

7 Mathematical Problems in Engineering Special Issue on Modeling Experimental Nonlinear Dynamics and Chaotic Scenarios Call for Papers Thinking about nonlinearity in engineering areas, up to the 70s, was focused on intentionally built nonlinear parts in order to improve the operational characteristics of a device or system. Keying, saturation, hysteretic phenomena, and dead zones were added to existing devices increasing their behavior diversity and precision. In this context, an intrinsic nonlinearity was treated just as a linear approximation, around equilibrium points. Inspired on the rediscovering of the richness of nonlinear and chaotic phenomena, engineers started using analytical tools from Qualitative Theory of Differential Equations, allowing more precise analysis and synthesis, in order to produce new vital products and services. Bifurcation theory, dynamical systems and chaos started to be part of the mandatory set of tools for design engineers. This proposed special edition of the Mathematical Problems in Engineering aims to provide a picture of the importance of the bifurcation theory, relating it with nonlinear and chaotic dynamics for natural and engineered systems. Ideas of how this dynamics can be captured through precisely tailored real and numerical experiments and understanding by the combination of specific tools that associate dynamical system theory and geometric tools in a very clever, sophisticated, and at the same time simple and unique analytical environment are the subject of this issue, allowing new methods to design high-precision devices and equipment. Authors should follow the Mathematical Problems in Engineering manuscript format described at Prospective authors should submit an electronic copy of their complete manuscript through the journal Manuscript Tracking System at mts.hindawi.com/ according to the following timetable: Guest Editors José Roberto Castilho Piqueira, Telecommunication and Control Engineering Department, Polytechnic School, The University of São Paulo, São Paulo, Brazil; piqueira@lac.usp.br Elbert E. Neher Macau, Laboratório Associado de Matemática Aplicada e Computação (LAC), Instituto Nacional de Pesquisas Espaciais (INPE), São Josè dos Campos, São Paulo, Brazil ; elbert@lac.inpe.br Celso Grebogi, Center for Applied Dynamics Research, King s College, University of Aberdeen, Aberdeen AB24 3UE, UK; grebogi@abdn.ac.uk Manuscript Due December 1, 2008 First Round of Reviews March 1, 2009 Publication Date June 1, 2009 Hindawi Publishing Corporation