To study the different classes and inequalities involving multivalent functions

Introduction and Definitions

Let p ∈ N {1, 2, 3,..} and (p) denote the class of functions f(z) of the form A function f(z) ∈ (p) is said to be p-valently starlike in , if it satisfies the inequality:

A function f(z) ∈ (p) is said to be p-valently convex in , if it satisfies the inequality:

Further, a function ( ) ∈ ( ) is said to p-valently close-to-convex in , if it satisfies the inequality:

(See, for details, [3] , [5] , and [13] for the above definitions.)

The following definitions of fractional calculus will be required in our present investigation:

Definition 1: (cf., [10] and [12] ; see also [2] ) Let a function f(z) be analytic in a simply-connected region of the z-plane containing the origin. The fractional integral of order ( > 0) is defined by

and the fractional derivative of order (0 ≤ < 1)is defined by

where the multiplicity of ( -) involved in (1.5) and that of ( -) in (1.6) are removed by requiring log ( -) to be real when -> 0.

Definition 2: (cf., [10] and [12] ; see also [2] ) Under the hypotheses of Definition 1, the fractional derivative of order + ( ∈ = ∪ {0}; 0 ≤ < 1) is defined by

Now, by making use of the fractional derivative operator , we define two important families ( ; ) and ( ; ) in ( ), where ∈ ℛ{0}, ∈ and 0 ≤ < 1.

Definition 3: Let ∈ ℛ{0}, ∈ and 0 ≤ 1. Then a function ( ) ∈ ( ) is said to belong to ( ; )if it satisfies the inequality:

where the value of ( )/ ( ) is taken its principal value.

by taking the principal value for ( ) .

Note that functions in ( ; 0) are p-valently starlike in (e.g. [9] ). See, for examples, the papers involving the fractional calculus and/or certain inequalities, [1] , [4] , [6] , [7] , and [11] . In [6,7] J, Irmak and cetin studied starlikeness and convexity for multivalent functions involving inequalities. In this paper we investigate various interesting properties for ( ; ) and ( ; ) associated with fractional calculus and also extend the results of Irmak and cetin ( [6,7] ).

Main Results

Now, we mention the following result which is used in the sequal.

Making use of this lemma, we first give the following theorem:

Theorem 1: Let ∈ ℛ{0}, ∈ and 0 ≤ < 1. If a function ( ) ∈ ( ) satisfies the inequality:

Proof: First of all, Definition 1 readily provides us the following fractional derivative formula for a power function:

Define the function w(z) by

Then it follows from (2.3) that w(z) is an analytic function in and w(0) = 0. The logarithmically differentiation of (2.4) implies that

Now, suppose that there exists a point ∈ such that

Then, applying Jack's Lemma, we can write

and ( ) = ( ≠ ). Thus, from (2.5) we obtain

where ≠ and ≥ 1. Therefore, (2.6) contradict our condition (2.2), and we conclude from the definition (2.4) that

which completes the proof of Theorem 1.

Theorem 2:2 Let ∈ ℛ{0}, ∈ and 0 ≤ < 1. If a funtion ( ) ∈ ( )satisfies the inequality:

then, using the same technique as in the proof of Theorem 1, we get the desired result. Many interesting results involving analytic and multivalent functions can be obtained by the use of Theorem 1 and Theorem 2 together with definitions (1.8) and (1.9) (respectivelly) and by choosing suitable values of , and p. Now, we are giving some of the important results for the analytic and geometric function theory (cf., [13]): Letting = 1 in Theorem 1, we have

References

1. Chen MP, Irmak H, Srivastava HM. Some families multivalently analytic functions with negative coefficients, J. Math. Anal. Appl. 1997; 214:674-690.
2. Choi JH. Some properties of multivalent functions associated with fractional calculus, Math. Japon. 50, 451-462, 1999.
3. Duren PL. Univalent Functions, Grundlehren der Mathematischen Wissenschaften 259, Springer-Verlag, New York, Berlin, Heidelberg, and Tokyo, 1983.
4. Dziok J, Srivastava HM. Classes of analytic functions associated with the generalized hypergeometric function, Appl. Math. Comput. 1999; 103:1-13.
5. Goodman AW. Univalent Functions, Vols. I and II, Polygonal Publishing House, Washington, New Jersey, 1983.
6. Irmak H, cetin 0F. Some theorems involving inequalities on p-valent functions, Turkish J Math. 1999; 23:453-459.
7. Irmak, H. and çetin, O. F., Some inequalities on p-valently starlike and pvalently convex functions, Hacettepe Bull. Natur. Sci. Engrg. Ser. B. 1999; 28:71-76.
8. Jack IS. Functions starlike and convex of order , J London Math. Soc. 1971; 3:469-474.
9. Miller SS, Mocanu PT. Second order differential inequalities in the complex plane, J Math. Anal. Appl. 1978; 65:289-305.
10. Owa S. On the distortion theorems. I, Kyungpook Math. J. 1978; 18:53-59.
11. Owa S, Kwon OS, Cho NE. Some inequalities involving meromorphically multivalent functions, J Math. Anal. Appl. 1997; 212:375-380.
12. Srivastava HM, Owa S. (Editors), Univalent Functions, Fractional Calculus, and Their Applications, Halsted Press (Ellis Horwood Limited, Chichester), John Wiley and Sons, New York, Chischester, Brisbane, and Toronto, 1989.
13. Srivastava HM, Owa S. (Editors), Current Topics in Analytic Function Theory, World Scientific Publishing Company, Singapore, New Jersey, London, and Hong Kong, 1992.