# On certain subclass of [lambda]-Bazilevic functions of type [alpha] + i[mu].

Abstract

In the present paper, the authors introduce a new subclass [B.sub.n]([lambda], [alpha], [mu], A, B, z) of [lambda]-Bazilevic functions of type [alpha] + i[mu]. The subordination relations and inequality properties are discussed by making use of differential subordination method. The results presented here generalize and improve some known results, and some other new results are obtained.

Keywords and Phrases: [lambda]-Bazilevic functions of type [alpha] + i[mu], Differential subordination.

1. Introduction and Definitions

Let [A.sub.n] denote the class of functions of the form

f(z) = z + [[infinity].summation over (k=n+1)] [a.sub.k][z.sup.k] (n [member of] N = {1, 2, 3,...}), (1.1)

which are analytic in the unit disk U = {z : z < 1}. Also let [S*.sub.n]([beta]) denote the usual class of starlike functions of order [beta], 0 [less than or equal to] [beta] < 1.

Let f(z) and F(z) be analytic in U. Then we say that the function f(z) is subordinate to F(z) in U, if there exists an analytic function [omega](z) in U such that [omega](z) [less than or equal to] z and f(z) = F([omega](z)), denoted by f [<] F or f(z) [<] F(z). If F(z) is univalent in U, then the subordination is equivalent to f(0) = F(0) and f(U) [subset] F(U) (see ).

The following class of analytic functions were studied by various authors (see ).

Definition 1. Let [B.sub.n]([alpha], [mu], [beta], g(z)) denote the class of functions in [A.sub.n] satisfying the inequality

R{[zf'(z)/f(z)] (f(z)/g(z))[.sup.[alpha]+i[mu]]} > [beta] (z [member of] U), (1.2)

where [alpha] [greater than or equal to] 0, [mu] [member of] R, 0 [less than or equal to] < [beta] < 1 and g(z) [member of] [S*.sub.n]([beta]). The function f(z) in this class is said to be Bazilevic function of type [alpha] + i[mu] and of order [beta].

In the present paper, we define the following class of analytic functions.

Definition 2. Let [B.sub.n]([lambda], [alpha], [mu], A, B, g(z)) denote the class of functions in [A.sub.n] satisfying the inequality

(1 - [lambda])[zf'(z)/f(z)] (f(z)/g(z))[.sup.[alpha]+i[mu]] + [lambda](1 + [zf"(z)/f'(z)]) (f'(z)/g'(z))[.sup.[alpha]+i[mu]] [<] [[1 + Az]/[1 + Bz]] (z [member of] U), (1.3)

where 0 [less than or equal to] [lambda] [less than or equal to] 1, [alpha] [greater than or equal to] 0, [mu] [member of] R, -1 [less than or equal to] B [less than or equal to] 1, A [not equal to] B, A [member of] R and g(z) [member of] [S*.sub.n]([beta]). All the powers in (1.3) are principal values, below we apply this agreement. The function f(z) in this class is said to be [lambda]-Bazilevic function of type [alpha]+i[mu].

If [alpha] = 1, [mu] = 0, A = 1-2[beta] and B = -1, then the class [B.sub.n]([lambda], [alpha], [mu], A, B, g(z)) reduces to the class of [lambda]-close-to-convex functions of order [beta], 0 [less than or equal to] [beta] < 1. If [alpha] = 0, [mu] = 0, A = 1 and B = -1, then the class [B.sub.n]([lambda], [alpha], [mu], A, B, g(z)) reduces to the class of [lambda]-convex functions . If [alpha] = 0, [mu] = 0, A = 1 - 2[beta] and B = -1, then the class [B.sub.n]([lambda], [alpha], [mu], A, B, g(z)) reduces to the class of [lambda]-convex functions of order [beta], 0 [less than or equal to] [beta] < 1. If A = 1 - 2[beta] and B = -1, then the class [B.sub.n]([lambda], [alpha], [mu], A, B, g(z)) reduces to the class of [lambda]-Bazilevic functions of type [alpha] + i[mu] and of order [beta], 0 [less than or equal to] [beta] < 1.

Li , Owa , Owa and Nunokawa  discussed the related properties of the classes [B.sub.n]([lambda], [alpha], [mu], 1-2[beta], -1, z), [B.sub.n](0, [alpha], 0, 1-2[beta], -1, z) and [B.sub.n]([lambda], 1, 0, 1-2[alpha], -1, z), respectively. In the present paper, we will discuss the subordination relations and inequality properties of the class [B.sub.n]([lambda], [alpha], [mu], A, B, z). The results presented here generalize and improve some known results, and some other new results are obtained.

2. Preliminaries Results

In order to establish our main results, we shall require the following lemmas.

Lemma 1 (). Let F(z) = 1 + [b.sub.n][z.sup.n] + [b.sub.n+1][z.sup.n+1] + ... be analytic in U, h(z) be analytic and convex in U, h(0) = 1. If

F(z) + [1/c]zF'(z) [<] h(z), (2.1)

where c [not equal to] 0 and R c [greater than or equal to] 0, then

F(z) [<] [c/n][z.sup.-[c/n]] [[integral].sub.0.sup.z] [t.sup.[c/n]-1]h(t)dt [<] h(z),

and [c/n][z.sup.-[c/n]] [[integral].sub.0.sup.z] [t.sup.[c/n]-1]h(t)dt is the best dominant for (2.1).

Lemma 2 (). Let f(z) = [[summation].sub.k=1.sup.[infinity]] [a.sub.k][z.sup.k] be analytic in U, g(z) = [[summation].sub.k=1.sup.[infinity]] [b.sub.k][z.sup.k] be analytic and convex in U. If f(z) [<] g(z), then [a.sub.k] [less than or equal to] [b.sub.1], for k = 1, 2,....

Lemma 3. Let 0 [less than or equal to] [lambda] [less than or equal to] 1, [alpha] [great than or equal to] 0, [mu] [member of] R, [alpha] + i[mu] [not equal to] 0, -1 [less than or equal to] B [less than or equal to] 1, A [not equal to] B and A [member of] R. Then f(z) [member of] [B.sub.n]([lambda], [alpha], [mu], A, B, z) if and only if

q(z) + [1/[[alpha] + i[mu]]]zq'(z) [<] [1 + Az]/[1 + Bz], (2.2)

where q(z) = (1 - [lambda]) (f(z)/z)[.sup.[alpha]+i[mu]] + [lambda](f'(z))[.sup.[alpha]+i[mu]].

Proof. Let

(f(z)/z)[.sup.[alpha]+i[mu]] = m(z). (2.3)

Then, by taking the derivatives in the both sides of (2.3), we have

[zf'(z)/f(z)] (f(z)/z)[.sup.[alpha]+i[mu]] = m(z) + [z/[[alpha] + i[mu]]]m'(z),

that is,

[zf'(z)/f(z)] (f(z)/z)[.sup.[alpha]+i[mu]] = (f(z)/z)[.sup.[alpha]+i[mu]] + [z/[[alpha] + i[mu]]] ((f(z)/z)[.sup.[alpha]+i[mu]])'. (2.4)

Substituting f(z) by zf'(z) in (2.4), we have

(1 + [zf"(z)/f'(z)])(f'(z))[.sup.[alpha]+i[mu]] = (f'(z))[.sup.[alpha]+i[mu]] + [z/[[alpha] + i[mu]]] ((f'(z))[.sup.[alpha]+i[mu]])'. (2.5)

From equalities (2.4) and (2.5), we get

(1 - [lambda])[zf'(z)/f(z)](f(z)/z)[.sup.[alpha]+i[mu]] + [lambda] (1 + [zf"(z)/f'(z)]) (f'(z))[.sup.[alpha]+i[mu]] = [(1 - [lambda])(f(z)/z)[.sup.[alpha]+i[mu]] + [lambda](f'(z))[.sup.[alpha]+i[mu]]] + [z/[[alpha]+ i[mu]]] [(1 - [lambda])(f(z)/z)[.sup.[alpha]+i[mu]] + [lambda](f'(z))[.sup.[alpha]+i[mu]]]'. (2.6)

Now, suppose that f(z) [member of] [B.sub.n]([lambda], [alpha], [mu], A, B, z), and let

q(z) = (1 - [lambda])(f(z)/z)[.sup.[alpha]+i[mu]] + [lambda](f'(z))[.sup.[alpha]+i[mu]].

Thus, from the definition of [B.sub.n]([lambda], [alpha], [mu], A, B, z) and equality (2.6), we can get (2.2).

On the other hand, this deductive process can be converse. Therefore, the proof of Lemma 3 is complete.

3. Main Results and Their Proofs

Theorem 1. Let 0 [less than or equal to] [lambda] [less than or equal to] 1, [alpha] [greater than or equal to] 0, [mu] [member of] R, [alpha] + i[mu] [not equal to] 0, -1 [less than or equal to] B [less than or equal to] 1, A [not equal to] B and A [member of] R. If f(z) [member of] [B.sub.n]([lambda], [alpha], [mu], A, B, z), then

(1 - [lambda]) (f(z)/z)[.sup.[alpha]+i[mu]] + [lambda](f'(z))[.sup.[alpha]+i[mu]] [<] [[[alpha] + i[mu]]/n] [[integral].sub.0.sup.1] [[1 + Azu]/[1 + Bzu]][u.sup.[[[alpha]+i[mu]]/n] - 1] du [<] [1 + Az]/[1 + Bz].

Proof. First let q(z) = (1 - [lambda]) (f(z)/z)[.sup.[alpha]+i[mu]] + [lambda] (f'(z))[.sup.[alpha]+i[mu]], then q(z) = 1 + [b.sub.n][z.sup.n] + [b.sub.n+1][z.sup.n+1] + ... is analytic in U. Now, suppose that f(z) [member of] [B.sub.n]([lambda], [alpha], [mu], A, B, z), by Lemma 3, we know

q(z) + [1/[[alpha] + i[mu]]]zq'(z) [<] [1 + Az]/[1 + Bz].

It is obvious that h(z) (1 + Az)/(1 + Bz) is analytic and convex in U,h(0) 1. Since [alpha] + i[mu] [not equal to] 0 and [alpha] [greater than or equal to] 0, therefore it follows from Lemma 1 that

(1 - [lambda])(f(z)/z)[.sup.[alpha]+i[mu]] + [lambda] (f'(z))[.sup.[alpha]+i[mu]] [<] [[[alpha] + i[mu]]/n][z.sup.-[[[alpha]+i[mu]]/n]] [[integral].sub.0.sup.z] [t.sup.[[[alpha]+i[mu]]/n] - 1]h(t)dt

= [[[alpha] + i[mu]]/n] [[integral].sub.0.sup.1] [[1 + Azu]/[1 + Bzu]][u.sup.[[[alpha]+i[mu]]/n] - 1] du [<] [1 + Az]/[1 + Bz].

Corollary 1. Let 0 [less than or equal to] [lambda] [less than or equal to] 1, [alpha] [greater than or equal to] 0, [mu] [member of] R, [alpha] + i[mu] [not equal to] 0 and [beta] [not equal to] 1. If f(z) [member of] [A.sub.n] satisfies

(1 - [lambda])[zf'(z)/f(z)] (f(z)/z)[.sup.[alpha]+i[mu]]+[lambda] (1 + [zf"(z)/f'(z)]) (f'(z))[.sup.[alpha]+i[mu]] [<] [[1 + (1 - 2[beta])z]/[1 - z]] (z [member of] U),

then

(1 - [lambda]) (f(z)/z)[.sup.[alpha]+i[mu]] + [lambda](f'(z)) [.sup.[alpha]+i[mu]] [<] [[[alpha] + i[mu]]/n] [[integral].sub.0.sup.1] [[1 + (1 - 2[beta])zu]/[1 - zu]][u.sup.[[[alpha] + i[mu]]/n] - 1]du (z [member of] U), (3.1)

and (3.1) is equivalent to

(1 - [lambda]) (f(z)/z)[.sup.[alpha]+i[mu]] + [lambda] (f'(z))[.sup.[alpha]+i[mu]] [<] [beta]+[[(1 - [beta])([alpha] + i[mu])]/n] [[integral].sub.0.sup.1] [[1 + zu]/[1 - zu]][u.sup.[[[alpha] + i[mu]]/n] - 1]du (z [member of] U).

Theorem 2. Let 0 [less than or equal to] [lambda] [less than or equal to] 1, [alpha] [greater than or equal to] 0, [mu] [member of] R, [alpha] + i[mu] [not equal to] 0, -1 [less than or equal to] B [less than or equal to] 1, A [not equal to] B and A [member of] R. If f(z) [member of] [B.sub.n]([lambda], [alpha], [mu], A, B, z), then

[inf.[z[member of]U]] R {[[[alpha] + i[mu]]/n] [[integral].sub.0.sup.1] [[1 + Azu]/[1 + Bzu]][u.sup.[[[alpha] + i[mu]]/n] - 1]du} < R {(1 - [lambda]) (f(z)/z)[.sup.[alpha]+i[mu]] + [lambda] (f'(z))[.sup.[alpha]+i[mu]]} < [sup.[z[member of]U]] R {[[[alpha] + i[mu]]/n] [[integral].sub.0.sup.1] [[1 + Azu]/[1 + Bzu]][u.sup.[[[alpha] + i[mu]]/n] - 1]du}.

Proof. Suppose that f(z) [member of] [B.sub.n]([lambda], [alpha], [mu], A, B, z), from Theorem 1 we know

(1 - [lambda])(f(z)/z)[.sup.[alpha]+i[mu]] + [lambda] (f'(z))[.sup.[alpha]+i[mu]] [<] [[[alpha] + i[mu]]/n] [[integral].sub.0.sup.1] [[1 + Azu]/[1 + Bzu]][u.sup.[[[alpha] + i[mu]]/n] - 1]du.

Therefore it follows from the definition of the subordination that

R {(1 - [lambda]) (f(z)/z)[.sup.[alpha]+i[mu]] + [lambda] (f'(z))[.sup.[alpha]+i[mu]]} > [inf.[z[member of]U]] R {[[[alpha] + i[mu]]/n] [[integral].sub.0.sup.1] [[1 + Azu]/[1 + Bzu]][u.sup.[[[alpha] + i[mu]]/n] - 1]du},

and

R {(1 - [lambda]) (f(z)/z)[.sup.[alpha]+i[mu]] + [lambda] (f'(z))[.sup.[alpha]+i[mu]]} < [sup.[z[member of]U]] R {[[[alpha] + i[mu]]/n] [[integral].sub.0.sup.1] [[1 + Azu]/[1 + Bzu]][u.sup.[[[alpha] +i[mu]]/n] - 1]du}.

Corollary 2. Let 0 [less than or equal to] [lambda] [less than or equal to] 1, [alpha] [greater than or equal to] 0, [mu] [member of] R, [alpha] + i[mu] [not equal to] 0 and [beta] < 1. If f(z) [member of] [B.sub.n]([lambda], [alpha], [mu], 1 - 2[beta], -1, z), then

[beta] +(1 - [beta]) [inf.[z[member of]U]] R {[[[alpha] + i[mu]]/n] [[integral].sub.0.sup.1] [[1 + zu]/[1 - zu]][u.sup.[[[alpha] + i[mu]]/n] - 1]du} < R {(1 -[lambda]) (f(z)/z)[.sup.[alpha] + i[mu]] + [lambda](f'(z))[.sup.[alpha]+i[mu]]} < [beta] + (1 - [beta])[sup.[z[member of]U]] R {[[[alpha] + i[mu]]/n] [[integral].sub.0.sup.1] [[1 + zu]/[1 - zu]][u.sup.[[[alpha] + i[mu]]/n] - 1]du}.

Corollary 3. Let 0 [less than or equal to] [lambda] [less than or equal to] 1, [alpha] [greater than or equal to] 0, [mu] [member of] R, [alpha] + i[mu] [not equal to] 0 and [beta] > 1. If f(z) [member of] [A.sub.n] satisfies

R {(1 - [lambda])[zf'(z)/f(z)] (f(z)/z)[.sup.[alpha]+i[mu]] + [lambda] (1 + [zf"(z)/f(z)]) (f'(z))[.sup.[alpha]+i[mu]]} < [beta] (z [member of] U),

then

[beta] + (1 - [beta]) [sup.[z[member of]U]] R {[[[alpha] + i[mu]]/n] [[integral].sub.0.sup.1] [[1 + zu]/[1 - zu]][u.sup.[[[alpha] + i[mu]]/n] - 1]du} < R {(1 - [lambda]) (f(z)/z)[.sup.[alpha]+i[mu]] + [lambda](f'(z))[.sup.[alpha]+i[mu]]} < [beta] + (1 - [beta]) [inf.[z[member of]U]] R {[[[alpha] + i[mu]]/n] [[integral].sub.0.sup.1] [[1 + zu]/[1 - zu]][u.sup.[[[alpha] + i[mu]]/n] - 1]du}.

Theorem 3. Let 0 [less than or equal to] [lambda] [less than or equal to] 1, [alpha] [greater than or equal to] 0 and -1 [less than or equal to] B < A [less than or equal to] 1. If f(z) [member of] [B.sub.n]([lambda], [alpha], 0, A, B, z), then

[[alpha]/n] [[integral].sub.0.sup.1] [[1 - Au]/[1 - Bu]] [u.sup.[[alpha]/n]-1]du < R{(1 - [lambda]) (f(z)/z)[.sup.[alpha]] + [lambda] (f'(z))[.sup.[alpha]]} < [[alpha]/n] [[integral].sub.0.sup.1] [[1 + Au]/[1 + Bu]] [u.sup.[[alpha]/n]-1]du (z [member of] U), (3.2)

and inequality (3.2) is sharp, with the extremal function defined by

(1 - [lambda]) (f(z)/z)[.sup.[alpha]] + [lambda](f'(z))[.sup.[alpha]] = [[alpha]/n] [[integral].sub.0.sup.1] [[1 + Au[z.sup.n]]/[1 + Bu[z.sup.n]]][u.sup.[[alpha]/n]-1]du. (3.3)

Proof. Suppose that f(z) [member of] [B.sub.n]([lambda], [alpha], 0, A, B, z), from Theorem 1 we know

(1 - [lambda]) (f(z)/z)[.sup.[alpha]] + [lambda](f'(z))[.sup.[alpha]] [<] [[alpha]/n] [[integral].sub.0.sup.1] [[1 + Azu]/[1 + Bzu]][u.sup.[[alpha]/n]-1]du.

Therefore it follows from the definition of the subordination and A > B that

R {(1 - [lambda]) (f(z)/z)[.sup.[alpha]] + [lambda] (f'(z))[.sup.[alpha]]} < [sup.[z[member of]U]] < R {[[alpha]/n] [[integral].sub.0.sup.1] [[1 + Azu]/[1 + Bzu]][u.sup.[[alpha]/n]-1]du} [less than or equal to] [[alpha]/n] [[integral].sub.0.sup.1] [sup.[z[member of]U]] R {[1 + Azu]/[1 + Bzu]}[u.sup.[[alpha]/n]-1]du < [[alpha]/n] [[integral].sub.0.sup.1] [[1 + Au]/[1 + Bu]][u.sup.[[alpha]/n]-1]du,

and

R{(1 - [lambda]) (f(z)/z)[.sup.[alpha]] + [lambda] (f'(z))[.sup.[alpha]]} > [inf.[z[member of]U]] R {[[alpha]/n] [[integral].sub.0.sup.1] [[1 + Azu]/[1 + Bzu]][u.sup.[[alpha]/n]-1]du} [greater than or equal to] [[alpha]/n] [[integral].sub.0.sup.1] [inf.[z[member of]U]] R {[1 + Azu]/[1 + Bzu]} [u.sup.[[alpha]/n]-1]du > [[alpha]/n] [[integral].sub.0.sup.1] [[1 - Au]/[1 - Bu]][u.sup.[[alpha]/n]-1]du.

It is obvious that inequality (3.2) is sharp, with the extremal function defined by equality (3.3).

Corollary 4. Let 0 [less than or equal to] [lambda] [less than or equal to] 1, [alpha] [greater than or equal to] 0 and [beta] < 1. If f(z) [member of] [B.sub.n]([lambda], [alpha], 0, 1 - 2[beta], -1, z), then

[[alpha]/n] [[integral].sub.0.sup.1] [[1 - (1 - 2[beta])u]/[1 + u]][u.sup.[[alpha]/n]-1]du < R {(1 - [lambda]) (f(z)/z)[.sup.[alpha]] + [lambda](f'(z))[.sup.[alpha]]} < [[alpha]/n] [[integral].sub.0.sup.1] [[1 + (1 - 2[beta])u]/[1 - u]][u.sup.[[alpha]/n]-1]du (z [member of] U), (3.4)

and inequality (3.4) is equivalent to

[beta] + [[(1 - [beta])[alpha]]/n] [[integral].sub.0.sup.1] [[1 - u]/[1 + u]][u.sup.[[alpha]/n]-1]du < R {(1 - [lambda]) (f(z)/z)[.sup.[alpha]] + [lambda](f'(z))[.sup.[alpha]]} < [beta] + [[(1 - [beta])[alpha]]/n] [[integral].sub.0.sup.1] [[1 + u]/[1 - u]][u.sup.[[alpha]/n]-1]du (z [member of] U).

Corollary 5. Let [alpha] [greater than or equal to] 0 and [beta] < 1. If f(z) [member of] [A.sub.n] satisfies

R{(1 + [zf"(z)/f'(z)]) (f'(z))[.sup.[alpha]]} > [beta] (z [member of] U),

then

[[alpha]/n] [[integral].sub.0.sup.1] [[1 - (1 - 2[beta])u]/[1 + u]][u.sup.[[alpha]/n]-1]du < R{(f'(z))[.sup.[alpha]]} < [[alpha]/n] [[integral].sub.0.sup.1] [[1 + (1 - 2[beta])u]/[1 - u]][u.sup.[[alpha]/n]-1]du (z [member of] U), (3.5)

[beta] + [[(1 - [beta])[alpha]]/n] [[integral].sub.0.sup.1] [[1 - u]/[1 + u]][u.sup.[[alpha]/n]-1]du < R{(f'(z))[.sup.[alpha]]} < [beta] + [[(1 - [beta])[alpha]]/n] [[integral].sub.0.sup.1] [[1 + u]/[1 - u]][u.sup.[[alpha]/n]-1]du (z [member of] U),

inequality (3.5) is sharp, with the extremal function defined by

[f.sub.[alpha],[beta]](z) = [[integral].sub.0.sup.z] ([[alpha]/n] [[integral].sub.0.sup.1] [[1 + (1 - 2[beta])[t.sup.n]u]/[1 - [t.sup.n]u]][u.sup.[[alpha]/n]-1]du)[.sup.1/[alpha]]dt. (3.6)

Corollary 6. Let [lambda] > 0 and 0 [less than or equal to] [beta] < 1. If f(z) [member of] [B.sub.n]([lambda], 1, 0, 1 - 2[beta], -1, z), then for z = r < 1, we have

R{f'(z)} > [1/[lambda]] [[integral].sub.0.sup.1] [t.sup.[1/[lambda]]-1][[1 - (1 - 2[beta])t]/[1 + t]]dt.

Remark 1. Corollary 6 is the corresponding result obtained by Owa and Nunokawa in .

By applying the similar method as in Theorem 3, we have

Theorem 4. Let 0 [less than or equal to] [lambda] [less than or equal to] 1, [alpha] [greater than or equal to] 0 and -1 [less than or equal to] A < B [less than or equal to] 1. If f(z) [member of] [B.sub.n]([lambda], [alpha], 0, A, B, z), then

[[alpha]/n] [[integral].sub.0.sup.1] [[1 + Au]/[1 + Bu]][u.sup.[[alpha]/n]-1]du < R{(1 - [lambda]) (f(z)/z)[.sup.[alpha]] + [lambda](f'(z))[.sup.[alpha]]}

< [[alpha]/n] [[integral].sub.0.sup.1] [[1 - Au]/[1 - Bu]][u.sup.[[alpha]/n]-1]du (z [member of] U), (3.7)

and inequality (3.7) is sharp, with the extremal function defined by equality (3.3).

Corollary 7. Let 0 [less than or equal to] [lambda] [less than or equal to] 1, [alpha] [greater than or equal to] 0 and [beta] > 1. If f(z) [member of] [A.sub.n] satisfies

R{(1 - [lambda])[zf'(z)/f(z)] (f(z)/z)[.sup.[alpha]] + [lambda](1 + [zf"(z)/f'(z)]) (f'(z))[.sup.[alpha]]} < [beta] (z [member of] U),

then

[[alpha]/n] [[integral].sub.0.sup.1] [[1 + (1 - 2[beta])u]/[1 - u]][u.sup.[[alpha]/n]-1]du < R{(1 - [lambda])(f(z)/z)[.sup.[alpha]] + [lambda] (f'(z))[.sup.[alpha]]}

< [[alpha]/n] [[integral].sub.0.sup.1] [[1 - (1 - 2[beta])u]/[1 + u]][u.sup.[[alpha]/n]-1]du (z [member of] U), (3.8)

and inequality (3.8) is equivalent to

[beta] + [[(1 - [beta])[alpha]]/n] [[integral].sub.0.sup.1] [[1 + u]/[1 - u]][u.sup.[[alpha]/n]-1]du < R{(1 - [lambda]) (f(z)/z)[.sup.[alpha]] + [lambda](f'(z))[.sup.[alpha]]}

< [beta] + [[(1 - [beta])[alpha]]/n] [[integral].sub.0.sup.1] [[1 - u]/[1 + u]][u.sup.[[alpha]/n]-1]du (z [member of] U).

Corollary 8. Let [alpha] [greater than or equal to] 0 and [beta] > 1. If f(z) [member of] [A.sub.n] satisfies

R{(1 + [zf"(z)/f'(z)]) (f'(z))[.sup.[alpha]]} < [beta] (z [member of] U),

then

[[alpha]/n] [[integral].sub.0.sup.1] + [[1 + (1 - 2[beta])u]/[1 - u]][u.sup.[[alpha]/n]-1]du < R{(f'(z))[.sup.[alpha]]} < [[alpha]/n] [[integral].sub.0.sup.1] [[1 - (1 - 2[beta])u]/[1 + u]][u.sup.[[alpha]/n]-1]du (z [member of] U), (3.9)

and inequality (3.9) is equivalent to

[beta] + [[(1 - [beta])[alpha]]/n] [[integral].sub.0.sup.1] [[1 + u]/[1 - u]][u.sup.[[alpha]/n]-1]du < R{(f'(z))[.sup.[alpha]]} < [beta] + [[(1 - [beta])[alpha]]/n] [[integral].sub.0.sup.1] [[1 - u]/[1 + u]][u.sup.[[alpha]/n]-1]du (z [member of] U),

inequality (3.9) is sharp, with the extremal function defined by equality (3.6).

Remark 2. Corollary 7-8 improve the corresponding results of Corollary 6-7 in , respectively.

If R[omega] [greater than or equal to] 0, then (R[omega])[.sup.1/2] [less than or equal to] R[[omega].sup.1/2] [less than or equal to] [omega](z)[.sup.1/2] (see [2, 9]). So we have

Theorem 5. Let 0 [less than or equal to] [lambda] [less than or equal to] 1, [alpha] [greater than or equal to] 0 and -1 [less than or equal to] B < A [less than or equal to] 1. If f(z) [member of] [B.sub.n]([lambda], [alpha], 0, A, B, z), then

([[alpha]/n] [[integral].sub.0.sup.1] [[1-Au]/[1-Bu]][u.sup.[[alpha]/n]-1]du)[.sup.1/2] < R{[(1 - [lambda])(f(z)/z)[.sup.[alpha]] + [lambda](f'(z))[.sup.[alpha]]][.sup.1/2]}

< ([[alpha]/n] [[integral].sub.0.sup.1] [[1 + Au]/[1 + Bu]][u.sup.[[alpha]/n]-1]du)[.sup.1/2] (z [member of] U) (3.10)

and inequality (3.10) is sharp, with the extremal function defined by equality (3.3).

Proof. From Theorem 1 we know

(1 - [lambda])(f(z)/z)[.sup.[alpha]] + [lambda](f'(z))[.sup.[alpha]] [<] [[1 + Az]/[1 + Bz]].

Since -1 [less than or equal to] B < A [less tan or equal to] 1, we have

0 [less than or equal to] [1-A]/[1-B] < R{(1 - [lambda]) (f(z)/z)[.sup.[alpha]] + [lambda](f'(z))[.sup.[alpha]]} < [1 + A]/[1 + B].

Thus, from inequality (3.2), we can get inequality (3.10). It is obvious that inequality (3.10) is sharp, with the extremal function defined by equality (3.3).

By applying the similar method as in Theorem 5, we have

Theorem 6. Let 0 [less than or equal to] [lambda] [less than or equal to] 1, [alpha] [greater than or equal to] 0 and -1 [less than or equal to] A < B [less than or equal to] 1. If f(z) [member of] [B.sub.n]([lambda], [alpha], 0, A, B, z), then

([[alpha]/n] [[integral].sub.0.sup.1] [[1+Au]/[1+Bu]][u.sup.[[alpha]/n]-1]du)[.sup.1/2] < R{[(1-[lambda]) (f(z)/z)[.sup.[alpha]] + [lambda](f'(z))[.sup.[alpha]]][.sup.1/2]}

< ([[alpha]/n] [[integral].sub.0.sup.1] [[1-Au]/[1-Bu]][u.sup.[[alpha]/n]-1]du)[.sup.1/2] (z [member of] U), (3.11)

and inequality (3.11) is sharp, with the extremal function defined by equality (3.3).

Remark 3. From Theorem 5-6 we also can obtain the corresponding results about some other special classes of analytic functions, here we don't give unnecessary details any more.

Theorem 7. Let 0 [less than or equal to] [lambda] [less than or equal to] 1, [alpha] [greater than or equal to] 0, -1 [less than or equal to] B [less than or equal to] 1, A [not equal to] B and A [member of] R. If f(z) = z + [[summation].sub.k=n+1.sup.[infinity]] [a.sub.k][z.sup.k] [member of] [B.sub.n]([lambda], [alpha], 0, A, B, z), then

[[alpha].sub.n+1] [less than or equal to] [A - B]/[(n + [alpha])([lambda]n + 1)], (3.12)

and inequality (3.12) is sharp, with the extremal function defined by equality (3.3).

Proof. Suppose that f(z) = z + [[summation].sub.k=n+1.sup.[infinity]] [a.sub.k][z.sup.k] [member of] [B.sub.n]([lambda], [alpha], 0, A, B, z), then we have

(1 - [lambda])[zf'(z)/f(z)] (f(z)/z)[.sup.[alpha]] + [lambda](1 + [zf"(z)/f'(z)])(f'(z))[.sup.[alpha]]

= 1 + (n + [alpha])([lambda]n + 1)[a.sub.n+1][z.sup.n] + ... [<] [1 + Az]/[1 + Bz].

It follows from Lemma 2 that

(n + [alpha])([lambda]n + 1)[a.sub.n+1] [less than or equal to] A - B. (3.13)

Thus, from (3.13), we can get (3.12). Notice that

f(z) = z + [[A - B]/[(n + [alpha])([lambda]n + 1)]][z.sup.n+1] + ... [member of] [B.sub.n]([lambda], [alpha], 0, A, B, z),

we obtain that inequality (3.12) is sharp.

Acknowledgments

This work was supported by the Scientific Research Fund of Hunan Provincial Education Department and the Hunan Provincial Natural Science Foundation (No. 05JJ30013) of the People's Republic of China. The authors would like to thank Professor H. M. Srivastava for his careful reading of and constructive suggestions for the original manuscript.

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Zhi-Gang Wang ([dagger]), Chun-Yi Gao ([double dagger]), and Shao-Mou Yuan ([section])

College of Mathematics and Computing Science

Changsha University of Science and Technology

Changsha, Hunan, 410076, People's Republic of China

Received March 26, 2005, Accepted October 5, 2005.

*2000 Mathematics Subject Classification. Primary 30C45.

([dagger]) E-mail: zhigwang@163.com

([double dagger]) E-mail: cygao10@163.com

([section]) E-mail: shaomouyuan@163.com
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