Some Applications of Ordinary Differential Operator to Certain Multivalent Functions

The aim of this paper is to apply the well-known ordinary differential operator to certain multivalent functions which are analytic in the certain domains of the complex plane and then to determine some criteria concerning analytic and geometric properties of the related complex functions.


Introduction, Notations, and Definitions
Let A(; ) denote the class of functions () of the following form: which are analytic and multivalent in the domain where C is the set of complex numbers.As is known, the domains U and D are known as unit open disk and punctured open unit disk, respectively.Also let M() := A(−; ) and T() := A(; ) when  ∈ N.
By differentiating both sides of the function () in the form (1), -times with respect to complex variable , one can easily derive the following (ordinary) differential operator: where  ≥ ,  ∈ N, and  ∈ N 0 := N ∪ {0}.
In this investigation, by applying the differential operator, defined by (3), to certain analytic functions which are multivalent in U or meromorphic multivalent in D, several criteria, which also include both analytic and geometric properties of univalent functions (see [1,2]), for functions () in the classes M() and T(), are then determined.In the literature, by using certain operators, several researchers obtained some results concerning functions belonging to the general class A(; ).In this paper, we also determined many results which include starlikeness, convexity, close-toconvexity, and close-to-starlikeness of analytic functions in 2 Journal of Mathematics the second section of this paper.One may refer to some results determined by ordinary differential operator in [3,4], some properties of certain linear operators in [5][6][7], and also certain results appertaining to multivalent functions and some of their geometric and analytic properties in [8,9] in the references.
For the proofs of the main results, we then need to recall the well-known method which was obtained by Jack [10] (see also [11]) and given by the following lemma.

Lemma 1. Let the function 𝑤(𝑧) given by
then where c is real number and  ≥  ≥ 1.

The Main Results and Their Applications
Theorem 2. Let () ∈ M() and  ∈ D, and also let the following inequality: be true.Then Proof.Let () ∈ M().By applying the differential operator, defined in (3), to the function (), one easily get that where (, ) and Γ(, ) are defined by Γ (, ) := ! ( − )! ( =  + 1,  + 2, . . .;  ≥ ;  ∈ N;  ∈ N 0 ) , (11) respectively.It is clear that the defined function () has the form in Lemma 1; that is, it is analytic in U with () ̸ ≡ 0. Upon differentiating of the identity (9), one easily obtains that Now suppose that there exists a point by applying Lemma 1; we then have  0   ( 0 ) = ( 0 ) ( ≥  ≥ 1).Thus, in view of the above equality, it can be calculated that which contradicts (7).Hence, we conclude that |()| < 1 for all  U, and Definition (9) yields the inequality which is equivalent to (8).Therefore, this completes the desired proof.Proof.Let the functions () ∈ M() and () ∈ M() be in the form respectively.Then, from related differential operator and definitions of the functions () and (), determine where (, ) and Γ(, ) are defined by ( 10) and ( 11), respectively.Define V() by Clearly, it is easily seen that V() satisfies the conditions of Lemma 1.The definition in (21) clearly gives us ) . ( Assume now that there exists a point  0 ∈ U such that max Then, applying Lemma 1, it follows from ( 16) and ( 22) that            (, ) which is contradicting to the assertion (17 It is obvious that () is analytic in U with (0) = 0.By differentiating () in (28) logarithmically, we find that and also suppose that there exists a point  0 ∈ U such that max