New classes of analytic and bi-univalent functions

which are analytic in the open unit disk U = {z ∈ C : |z| < 1} and normalized by the conditions f (0) = 0, f ′(0) = 1. Let S ⊂ A denote the class of all functions inA which are univalent in U. The Koebe One-Quarter Theorem [5] ensures that the image of the unit disk under every f ∈ S functions contains a disk of radius 1/4. It is well known that every functions f ∈ S has an inverse f −1, which is defined by f −1( f (z)) = z, z ∈ U

For the new results posted in the last section we have to recall the necessary elements of the (p, q)calculus involving.
There is possibility of extension of the q-calculus to post quantum calculus denoted by the (p, q)calculus.
When the case p = 1 in (p, q)-calculus, the q− calculus may be obtained. In order to derive our main results, we need to following lemmas: Lemma 1. [15] If p ∈ P, then |c k | ≤ 2 for each k, where P is the family of all functions p analytic in U for which Lemma 2.
From the relation (1.2), we deduce that [k] p,q a k z k−1 (1.5) where the (p, q)-bracket number or twin-basic is given by which is a natural generalization of the q-number. [4,16].

Main results
Definition 5. A function f given by (1.1) is said to be in the class H p,q,α , if the following conditions are satisfied: where the function g is given by (1.2).
. In the next theorem we obtain coefficient bounds for the functions class H p,q,α Σ .

Theorem 7.
Let the function f given by (1.1) be in the function class H p,q,α and Proof. From the relations (2.1) and (2.2) it follows that From the relation (2.5), we obtain the next relations It follows that The relations (2.10) and (2.11) are obtained from the relations (2.6) and (2.8). We obtain that This relation is obtained from (2.9) and (2.11). We get from (2.12) (2.13) From Lemma 1 for the above equality, we get the estimate on the coefficient |a 2 | as asserted in the relation (2.3). We subtract (2.9) from (2.7) and find the bound on the coefficient |a 3 |. We get it that way 14) It follows that, from the relations (2.10), (2.11) and (2.14). And if we apply Lemma 1 for the above equality, we obtain the estimate on the coefficient |a 3 | as asserted in (2.4).
In the next theorem we obtain coefficient bounds for the functions class H p,q,β Σ .
We obtain By using the relation (2.26) into (2.27) it follows that If we apply Lemma 1 for the relations (2.28) and (2.29) we get the estimate on the coefficient |a 3 | as asserted in (2.18).
We obtain the next corollary.
Corollary 12. Let f (z) given by (1.1) be in the function class H Σ (β)(0 ≤ β < 1). Then Definition 13. Let b, t : U → C be analytic functions and where z, w ∈ U and the function g is given by (1.2).
In the next theorem we obtain coefficient bounds for the functions class H p,q,b,t Σ .
Theorem 14. Let f given by (1.1) be in the class H p,q,b,t Σ . Then Proof. We will write the equivalent forms of the argument inequalities in the relations (2.32) and (2.33).
We find that [2] p,q a 2 = b 1 (2.38) It follows that Substituting the value of a 2 2 from the relation (2.46) into the relation (2.47), we get