ON COEFFICIENT BOUNDS OF A CERTAIN CLASS p-VALENT ,-SPIRAL FUNCTIONS OF ORDER

S ABSTRACT. Let (A,B,p,a)(II < -I & A < B & and o a < p), denote the n class of functions f(z) z p + ( a z analytic in U {z: Izl < I}, which n iO n=p+l satisfy for z re U

+ Bw(z)   w(z) is analytic in U with w(o)= o and lw(z) Izl for z U.In this paper we obtain the bounds of an and we maximize lap+2 a2p+l over the S l class (A,B,p,a) for complex values of KEY WORDS AND PHRASES.p-Valent, analytic, bounds, -spirallike functions of order 19BOAMS SUBJECT CLASSIFICATION CODE.30A32 1. INTRODUCTION.
Let A (p a fixed integer greater than zero) denote the class of functions P f(z) z p + akzk which are analytic in U {z: Izl < I}.We use to denote k=p+l the class of bounded analytic functions w(z) in U satisfies the conditions w(o)--o and lw(z) l-<-Izl for z e U. Also let e (p a) (with p a postive integer) denote the class of functions with positive real part of order a that have the form The class P(A,B) was introduced by Janowski [2].
For -I =< A < B and o =< a < p, denote by P(A,B,p,a) the class of func- tions P2(z) of form (I.I) which satisfy that P2(z) P(A,B,p,a) if and only if P2(z) (p-a)P l(z) + a, Pl(Z) E P(A,B) (1.6) Using (1.5) in (1.6), one can show that P2(z) E P(A,B,p,a) if and only if P2(z) p + [pB+(A-B) (p-a) ]w(z) + Bw(z)   (2.1) PROOF.We prove the lemma by induction on m.
Next suppose that the result is true for m=q-l.
For m=l the lemma is obvious.
We have cos2 ql  which is equivalent to (2.2).
To establish (2.2) for n > p+l, we will apply induction argument.(2.9) Thus from (2.7), (2.9) and lemma 2 with m--n-p, we obtain This completes the proof of (2.2).This proof is based on a technique found in Clunie [4].
For sharpness of (2. (2) Setting B=I, A=-I and p=l in Theorem I, we get the result of Libera [5].
(3) Setting B=I, A=-I, p=l and a=0 in Theorem I, we get the result of Zamorski [6].
Remark on Theorem 2. Setting (i) B=I and A=-I, (li) B=I, A=-I and p=l, (iii) B=I, A=-I, p=l and e=0, (iv) B=I, A=-I and =0, in Theorem 2, we get the results of Patil and Thakare [I].

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 http://www .hindawi.com/journals/mpe/.Prospective authors should submit an electronic copy of their complete manuscript through the journal Manuscript Tracking System at http:// mts.hindawi.com/according to the following timetable:

P
in U and satisfy the conditions P(o) p and Re{P(z)} > a (o --< a < p) in U.The class P(p,a) was introduced by Patil and Thakare [I].It was shown in [I] that the function P P(p,a) if and only if P(z) p-(p-2a)w(z) .We say the functions in S%(p,a) are p-valent l-spiral- $ like of order a The class (p,a) was introduced by Patil and Thakare [I].It was shown in [i] that f Sl(p,e) if and only if there exists a function p e P(p,a) the class of functions Pl(Z) + Z n=l analytic in U and such that Pl(Z) P(A,B) if and only if l+Aw A CERTAIN CLASS OF p-VALENT FUNCTIONS COEFFICIENT ESTIMATES FOR THE CLASS Sk(A,B,p,a).

First
Round of Reviews March 1, 2009