By , we denote the class of functions of the type:
which are p-valent and analytic in the open unit disk , see .
Now, we introduce some basic definitions and related details of the q-calculus, see   .
The q-shifted factorial is defined for as a product of n factors by:
and according to the basic analogue of the gamma function, we get:
where the q-gamma function is given by:
If the relation (2) is meaningful for as a convergent product defined by:
Further, we conclude that
For , the q-derivative of a function f is defined by:
A simple calculation yields that for and ,
Also, in view of the following relation:
we note that the q-shifted factorial (2) reduces to the well-known Pochhammer symbol , which is defined by:
Differentiating (1) m times with respect to z (8), we conclude
A function is said to be in the subclass if it satisfies the inequality:
where , , and . Indeed is said to be in the subclass if it satisfies the inequality:
For details see .
2. Main Results
To prove the main theorems related to and , we need the following lemma due to Jack  .
Lemma 1. Let e non-constant in and . If attains its maximum value on the circle at , then , where is a real number.
A function is said to be in the subclass of p-valently close-to-convex functions with respect to the origin in if
Also, is said to be in the subclass of p-valently starlike functions with respect to the origin in if
Further is said to be in the subclass of p-valently convex functions with respect to the origin in if
see  .
Theorem 2. If satisfies the inequality:
Proof. Let , we define the function by:
with a simple calculation we have (in ).
For (14), we obtain:
From (14) and (15), we get:
Now, let for , , then by using Jack’s lemma and putting in (16), we have:
which is a contradiction with (13). Thus we have for all , so from (14) we conclude:
and this gives the result.
By letting and ( ), we have the following corollaries which are due to Irmak and Cetin .
Corollary 3. If satisfies
Corollary 4. If satisfies the inequality
then and .
Theorem 5. If satisfies
Proof. Let the function , we define the function by
It is easy to verify that is analytic in and . By (18), we have:
or by (18) we get
Now, let for a point , . By Jack’s lemma and putting we conclude:
which is contradiction with (17). Thus for all , and so from (18), we have:
thus the proof is complete.
By letting and ( ) we have the following corollaries that the first one is due to Irmak and Cetin .
Corollary 6. If satisfies the inequality
then and .
Studying the theory of analytic functions has been an area of concern for many authors. Literature review indicates lots of researches on the classes of p-valent analytic functions. The interplay of geometric structures is a very important aspect in complex analysis. In this study, two new subclasses of p-valent functions were defined by using q-analogue of the well-known operators and we gave some geometric structures like starlike, convex and close-to-convex properties of the subclasses. It is noted that the study is an extension of some previous studies as it is shown in corollaries 3, 4, 6.
The authors wish to thank the reviewer for their valuable suggestions which add to the quality of this paper.
 El-Qadeem, A.H. and Mamon, M.A. (2018) Comprehensive Subclasses of Multivalent Functions with Negative Coefficients Defined by Using a q-Difference Operator. Transactions of A. Razmadze Mathematical Institute, 172, 510-526.
 Miller, S.S. and Mocanu, P.T. (1978) Second Order Differential Inequalities in the Complex Plane. Journal of Mathematical Analysis and Applications, 65, 289-305.