ON A CLASS OF p-VALENT CLOSE-TO-CONVEX FUNCTIONS OF ORDER / 3 AND TYPE

Let S(A,B,p, =) denote the class of functions g(z) z p + E b z n n n--p+l analytic in the unit disc U {z: Izl < I} and satisfying the condition


I. INTRODUCTION.
Let A (p a fixed integer greater than zero) denote the class of functions P z p + I akzk which are analytic in U {z: Izl < I}.Let denote the f(z) k=p+l class of bounded analytic functions w(z) in U satisfying the conditions w(o) o and lw(z) --< Izl, for z U. We use e to denote the class of functions P1 (z)   + l dn zn which are analytic in U and have a positive real part there.
n--I k Also we use P(p,8) the class of functions that have the form P(z) p + l c k z k=l which are analytic in U and satisfy the conditions P(o) p and Re{P(z)} > 8 (o-<-8 <p) in U.The class P(p,8) was introduced by  Obviously S(A,B,p,a) is a subclass of the class S (a), o a< p, of p-valent star- P like functions of order a, investigated by Goluzina  [2].The class S(A,B,p,a) introduced by the author [3].(1.3) Thus if f(z) e C(A,B,p,8,a), then we may write zf'(z) g(z) P(z), P(z) P(p,fl). (1.4) We note that C(l,-l,l,fl,) C(8,), is a subclass of the class of close-to- convex functions of order 8 and type c introduced by Libera [4]. (z) , (z) we shall mean the Hadamard product or convolution of (z) and (z), We state below some lemmas that are needed in our investigation.shows that the result is sharp for each n > I. shows that the result is sharp for each n p+l.This gives nlanl--< Plbnl + IClllbn_ll + Ic211bn_21 + + ICn_p_lllbp+ll + Substituting the value from (1.5) and (1.8) in (2.4), we obtain (2.4) which, on simplification takes the form of (2 I) The function f (z) defined by O f'(z) z p-I P+(P-2B)IZ shows that the bound (2.1) is sharp for each n > p+l.Remarks on Theorem I. (I) Choosing p=l and =B=o in Theorem I, we get the result due to Goel and Mehrok [6].
M.K. AOUF (2) Choosing p=l, A=I, B=-I and e==o in Theorem I, we get the result due to Reade [7].sPg(tz) PROOF.The proof is similar to the one given by Ruscheweyh [8] and Goel and Mehrak [6].

P(z) (p 8 )
P1(z) + 8 (i.I) bor -i <--B < A-<- and o < p, denote by S(A,B,p,e) the class of functions gsubordination it follows that g(z) S(A,B,p,a) has a representa- tion of the form zg'(z) p+[pB + (A-B) (p-a) ]w(z) Moreover, let C(A,B,p,B,a) denote the class of functions f(z) A which P satisfy Re {z_(z)} > 8 g e S(A,B p a) g(z)
(-A(p-e)r) < Ig(z)I < r p exp(A(p-x)r), B o. B) (p-a) dr rP_l p!Ip-2Slrl_r (l+Br) B dr B # o', o r 2B f r p-I P-IPexp(-A(p-a)r)dr < If(z)] p+lp-2Bl r exp (A (p-)r) dr, B o. l-r Let z z ol r be chosen in such a way that f(z o)1 < If(z) for all z O r.If L(z is the pre-image of the segment [o f(Zo)] in U (= )I f 17<z)lld-I l(z) Idr--> o L(z L(z o o ; rP-1 p_-lp-2Blr l+r o exp(-A(p-e)r)dr, B o.
Patil and Thakare [i].It is well known that a function P(z) is in P(p,8) if and only if there exists a function Pl(Z) e P such that