A note on boundedness properties of Wright’s generalized hypergeometric functions

. In this paper we obtain some inequalities giving the boundedness properties for the Wright’s generalized hypergeometric function which belong to the classes P(A,B) and R(A,B). The results besides yielding the inequalities obtained recently in [3] and [7], would also be applicable to special functions like, the Bessel-Maitland functions and Mittag-Leffler functions..

The symbol -is the usual symbol of subordination.That is for f(z), g(z) E S, f(z) ~ g(z) if there exists a Schwarz function w(z) (w(0)=0 , j w(z) ) I 1 in U) such that f(z) = g(w(z)).
We denote by P(A,B) the set of functions if h(z) ~ 1 + AZ , where A and B are real numbers such that -1 _ B A -l; -1 Also, R(A,B) denotes the set of functions If A= 1-2 a, B= -1, then the subclass of functions of S are denoted by P* (a).
Our purpose in this paper is to obtain some inequalities for the function defined by (I.I) which belong to the classes P(A,B) and R(A,B) .The boundedness properties for the generalized hypergeometric function as well as similar properties for the Bessel-Maitland and Mittag-Leffler functions follow as worthwhile consequences of our main inequalities.
Further Inequalties Theorem 2. Let A ~ o , and °A j~' J ~~~ ~ § z P(A,B) and I z I ~ ) I B I , then where 0394 is given by (2.1).
Proof.Since it follows then that 0394 d dz p w [ ' i , Ai > i , p 1 z = 0394p y [ h i + Ai, Ai) i , p ° z. (3.2) We recall the following result ([8]): If h(z If we set h(z) = 0394p y ( h I ' Ai) I , p ; z in (3 3)   i ( F i , I Bi) i , q i ' ° ' and use (3 .2) in the process, we are lead to the result (3 .I).

4.
Some Consequences of Theorems 1-3 By specializing the parameters, we observe that for Aj = 1 (j=l,...,p) and Bj =1 (j =l,...,q), the Wright's generalized hypergeometric function Setting the parameters of the Wright's generalized hypergeometric function occurring in Theorems 1-2 in accordance with (4.1), we get the results obtained recently in [3] (Ths.1-2, pp.67-70).In addition to the choice of parameters indicated above for (4.1), if we also put x=l in Theorem 3, we are then lead to the other known result [3] (Th. 3, p. 71); see also [7].Further, Theorems 1-2 can be applied to special functions like the Bessel-Maitland and Mittag-Leffler functions, and boundedness properties for these functions can be obtained.Indeed, by noting the relationships [7, p. where J 03BD (z) and F03B1, 03B2(z) are the Bessel-Maitland and Mittag -Leffler functions, respectively, the corresponding relations can easily be deduced from the main results.