Some Properties of Certain Subclasses of Analytic Functions with Complex Order

The main purpose of this paper is to derive some coefficient inequalities and subordination properties for certain subclasses of analytic functions involving the Salagean operator. Relevant connections of the results presented here with those obtained in earlier works are also pointed out.


Introduction
For 0 α < 1, we denote by S * α and K α the usual subclasses of A consisting of functions which are, respectively, starlike of order α and convex of order α in U. Clearly, we know that f ∈ K α ⇐⇒ zf ∈ S * α .We note that Given two functions f, g ∈ A, where f is given by 1.1 and g is defined by the Hadamard product or convolution f * g is defined by For two functions f and g, analytic in U, we say that the function f is subordinate to g in U, and write if there exists a Schwarz function ω, which is analytic in U with Indeed, it is known that Furthermore, if the function g is univalent in U, then we have the following equivalence: In recent years, Deng 6 see also Kamali where It is easy to see that the class S n λ, α, b includes the classes S * α and K α as its special cases.Now, motivated essentially by the above-mentioned function classes, we introduce the following subclass of A of analytic functions.Definition 1.2.A function f ∈ A is said to be in the class M n λ, β, b if it satisfies the inequality: where 1.17 It is also easy to see that the classes M β and N β are special cases of the class M n λ, β, b .
In this paper, we aim at proving some coefficient inequalities and subordination properties for the classes S n λ, β, b and M n λ, β, b .The results presented here would provide extensions of those given in earlier works.Several other new results are also obtained.

Coefficient Inequalities
In this section, we derive some coefficient inequalities for the classes S n λ, α, b and By noting that it follows from 2.2 that the above last expression is bounded by |b| 1 − α .This completes the proof of Theorem 2.1.

2.7
We consider M ∈ R defined by

2.9
It follows from 2.6 that M < 0, which implies that 2.7 holds, that is, f ∈ M n λ, β, b .The proof of Theorem 2.2 is evidently completed.

ISRN Mathematical Analysis
To prove our next result, we need the following lemma.
Lemma 2.3.Let β > 1 and b ∈ C \ {0}.Suppose also that the sequence {B j } ∞ j 1 is defined by Proof.We make use of the principle of mathematical induction to prove the assertion 2.11 of Lemma 2.3.Indeed, from 2.10 , we know that 12 which implies that 2.11 holds for j 2. We now suppose that 2.11 holds for j m m 2 , then

2.13
Combining 2.10 and 2.13 , we find that 14 which shows that 2.11 holds for j m 1.The proof of Lemma 2.3 is evidently completed.
Theorem 2.4.Let f ∈ M n λ, β, b , then Proof.We first suppose that where B j j n 1 − λ λj a j .

2.17
Next, by setting

2.18
we easily find that h ∈ P. It follows from 2.18 that zF z

2.19
We now find from 2.16 , 2.18 , and 2.19 that

2.20
By evaluating the coefficients of z j in both the sides of 2.20 , we get

2.21
On the other hand, it is well known that Combining 2.21 and 2.22 , we easily get

Subordination Properties
In view of Theorems 2.1 and 2.2, we now introduce the following subclasses: which consist of functions f ∈ A whose Taylor-Maclaurin coefficients satisfy the inequalities 2.2 and 2.6 , respectively.A sequence {b j } ∞ j 1 of complex numbers is said to be a subordinating factor sequence if, whenever f of the form 1.1 is analytic, univalent, and convex in U, we have the subordination To derive the subordination properties for the classes S n λ, α, b and M n λ, α, b , we need the following lemma.
where, for convenience, The constant factor in the subordination result 3.4 cannot be replaced by a larger one.

ISRN Mathematical Analysis
Proof.Let f ∈ S n λ, α, b and suppose that where Φ n, λ, α, b is defined by 3.7 .If is a subordinating factor sequence with a 1 1, then the subordination result 3.4 holds.By Lemma 3.1, we know that this is equivalent to the inequality is an increasing function of j, and using Theorem 2.1, we have

3.14
This evidently proves the inequality 3.12 , and hence also the subordination result 3.4 , asserted by Theorem 3.2.The inequality 3.5 asserted by Theorem 3.2 follows from 3.4 by setting

3.18
We thus complete the proof of Theorem 3.2.
The proof of the following subordination result is much akin to that of Theorem 3.2.We, therefore, choose to omit the analogous details involved.

Let A denote the class of functions of the form f z z ∞ j 2 a j z j , 1 . 1 which
are analytic in the open unit disk U : {z : z ∈ C, |z| < 1}.1.2