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On Certain Conditions of Geometric Functions for the Generalized Hypergeometric Functions

Chena Ram¹ and Garima Agarwal²

²Research Scholar, Department of Mathematics and Statistics, Jai Narain Vyas University, Jodhpur, Rajasthan, INDIA.

Corresponding Addresses:

¹[email protected], ²[email protected]

1. Introduction

   Let   denote the class consisting of functions  given by

                                                                                               (1.1)

which are analytic and univalent in the open unit disk U = {

   Let  is the subclass of the class  Which has functions satisfying the condition

                                                                                                      (1.2)

for some  and for all

      Also let  denote the subclass of  constituted by functions those satisfy the condition

                                                                                                            (1.3)

for some  and for all

    From (1.2) and (1.3), we have

                 .                                                                   (1.4)

 We note that   the class of starlike functions of order

and , the class of convex functions of order                       (see Silverman[5]).

    Following definitions [3] will be required in proving the main results:

₃₃R₂(               (1.5)

where
    If we set , then the generalized hypergeometric function ₃ reduces to the result due to Virchenko, Kalla and Zamel [6], as

₂₂R₁(.                                         (1.6)

   For  (1.5) reduces to the Clausenian hypergeometric function studied by Saxena and Kalla [4]

₃F₂(,                                                              (1.7)

and for , (1.6) reduces to the Gauss hypergeometric function

₂F₁(                                                                             (1.8)

    In the present paper we shall use the following lemmas, see Altintas and Owa [1].

Lemma 1. A function  of the form (1.1) is in the class , if and only if

) (1.9)

 Lemma 2. A function  of the form (1.1) is in the class , if and only if

)                                                            (1.10) 

Main Results

Theorem 2.1. If   then  ₃R₂( is in the class , if and only if

₃R₂(

₃R₂(            (2.1)

Proof: Since

 ₃R₂(

                                ,              (2.2)

by Lemma 1, it is sufficient to prove that

  )
Now we have

  )

                  (1                                                                                              

i.e.           (1                                                                                                              

i.e            (1                                                                                                                                      +,

which on using (1.5) yields

₃R₂(

₃R₂( 

The theorem is completely proved.    

Corollary 1.   ₃F₂( is in the class , if and only if

               ₃F₂(

₃F₂( (2.3)

 Proof. If we set =1 in (2.1), the proof is completed.

Corollary 2.   ₂R₁( is in the class , if and only if

               ₂R₁(

₂R₁(  (2.3)

Proof. If we set  in (2.1), the proof is completed.

Theorem 2.2. If  ), then

R₂(  [2 − ₃R₂( is in the class , if and only if

₃R₂(

₃R₂(                  (2.5)

Proof.

R₂(          (2.6)

By Lemma 1, we need only to show that         

Now,

=

i.e.          =

=

Hence,

₃R₂(

₃R₂( .       

The proof is completed. 

Corollary 3. F₂(  [2- ₃F₂( is in the class  if and only if

₃F₂(

₃F₂(                                                  (2.7)

Proof. If we set  in (2.5), the proof is completed.                                                

Corollary 4. R₁(  [2- ₂R₁( is in the class  if and only if

₂R₁(

₂R₁(                                       (2.8)

Proof. If we set  in (2.5), the proof is completed.                                                

Theorem 2.3. If   then  ₃R₂( is in the class , if and only if

₃R₂(

₃R₂(

                                  ₃R₂(                             (2.9)Proof. From (2.2), we have

 ₃R₂(

By Lemma 2, it is sufficient to prove that

)

Now we have

                                + 

                                                + 

+ 

                                                + 

                                + 

                                                + 

                                + 

                + 

=₃R₂(               

₃R₂(   ₃R₂(           

The proof is completed.