On Certain Sufficient Condition Involving Gaussian Hypergeometric Functions

Recommended by Teodor Bulboac˘ a The authors define a new subclass of A of functions involving complex order in the open unit disk U. For this new class, we obtain certain inclusion properties involving the Gaussian hypergeometric functions.


Introduction and Motivation
Let A be the class of functions f normalized by a n z n , 1.1 which are analytic in the open unit disk As usual, we denote by S the subclass of A consisting of functions which are also univalent in U. A function f ∈ A is said to be starlike of order α in U 0 ≤ α < 1 , if and only if This function class is denoted by S * α .We also write S * 0 : S * , where S * denotes the class of functions f ∈ A that are starlike in U with respect to the origin.

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A function f ∈ A is said to be convex of order α in U 0 ≤ α < 1 if and only if The class of convex functions is denoted by the class K α .Further, K K 0 , the well-known standard class of convex functions.It is an established fact that f ∈ K α ⇐⇒ zf ∈ S * α . 1.5 A function f ∈ A is said to be in the class UCV of uniformly convex functions in U if f is a normalized convex function in U and has the property that, for every circular arc δ contained in the unit disk U, with center ζ also in U, the image curve f δ is a convex arc.The function class UCV was introduced by Goodman 1 .
For functions f ∈ A given by 1.1 and g ∈ A given by g z z ∞ n 2 b n z n , we define the Hadamard product or Convolution of f and g by a n b n z n , z ∈ U.

1.6
Furthermore, we denote by k − UCV and k − ST two interesting subclasses of S consisting, respectively, of functions which are k-uniformly convex and k-starlike in U. Thus, we have

1.7
The class k − UCV was introduced by Kanas and Wiśniowska 2 , where its geometric definition and connections with the conic domains were considered.The class k − ST was investigated in 3 .In fact, it is related to the class k − UCV by means of the well-known Alexander equivalence between the usual classes of convex and starlike functions; see also the work of Kanas and Srivastava 4 for further developments involving each of the classes k − UCV and k − ST .In particular, when k 1, we obtain where UCV and SP are the familiar classes of uniformly convex functions and parabolic starlike functions in U, respectively see for details, 1, 5 .In fact, by making use of a certain fractional calculus operator, Srivastava and Mishra 6 presented a systematic and unified study of the classes UCV and SP .
A function f ∈ A is said to be in the class P τ γ A, B ⊂ A if it satisfies the inequality 1.9 The class P τ 0 A, B was introduced earlier by Dixit and Pal 7 .Two of the many interesting subclasses of the class P τ γ A, B are worthy of mention here.First of all, by setting Secondly, if we put we obtain the class of functions f ∈ A satisfying the inequality which was studied by among others Padmanabhan 9 and Caplinger and Causey 10 .Finally, many of the authors have also studied the class P 1 γ A, B .For details of these works one can refer to the works of Ding Gong 11 , R. Singh and S. Singh 12 , Owa and Wu 13 , and also the references cited by them.Although, many mapping properties of the class P 1 γ A, B have been studied by these authors, they did not study any mapping properties involving the hypergeometric functions.
The Gaussian hypergeometric function F a, b; c, z , z ∈ U is given by is the solution of the homogeneous hypergeometric differential equation and has rich applications in various fields such as conformal mappings, quasiconformal theory, and continued fractions.

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Here, a, b, c are complex numbers such that c / 0, −1, −2, −3, . .., a 0 1 for a / 0, and for each positive integer n, a n a a 1 a 2 • • • a n − 1 is the Pochhammer symbol.In the case of c −k, k 0, 1, 2, . .., F a, b; c; z is defined if a −j or b −j, where j ≤ k.In this situation, F a, b; c; z becomes a polynomial of degree j in z.In particular, the close-to-convexity in turn the univalency , convexity, starlikeness, for details on these technical terms we refer to 5 , and various other properties of these hypergeometric functions were examined based on the conditions on a, b, and c in 8 .For more interesting properties of hypergeometric functions, one can also refer to 20, 21 .
Let f z and g z be analytic in U and g z univalent.Then we say that f z is subordinate to g z written as f z ≺ g z if f 0 g 0 and f U ⊂ g U .For f ∈ A, we recall that the operator I a,b,c f of Hohlov 22 which maps A into itself defined by where * denotes usual Hadamard product of power series.Therefore, for a function f defined by 1.1 , we have 1.17 Using the integral representation, we can write Indeed, I 1,1,2 f and I 1,2,3 f are known as Alexander and Libera operators, respectively.
Let 0 ≤ k < ∞, and let f ∈ A be of the form 1.1 .If f ∈ k − UCV , then the following coefficient inequalities hold true cf. 2 : where which is the extremal function for the class P p k related to the class k − UCV by the range of the expression where P 1 P 1 k is given, as above, by 1.22 .
Similarly, if f of the form 1.1 belong to the class k − ST , then cf. 3 where P 1 P 1 k is given, as above by 1.22 .

Properties of P τ γ A, B
Theorem 2.1.Let f ∈ S and be of the form The estimate is sharp.
Proof.Since f ∈ P τ γ A, B , we have where w z is analytic in U and satisfies the condition w 0 0 and |w z | < 1 for z ∈ U. Hence, we have

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Using By equating the coefficients, we observe that the coefficient a n in the right-hand side depends only on a 2 , a 3 , . . ., a n−1 on the left-hand side of the above expression.This gives

2.7
By letting r → 1, we conclude that By making use of the fact that −1 ≤ B < 1, we get

2.11
The result is sharp for the function

2.13
The result is sharp for the function Proof.In view of 2.13 , which is clearly less than or equal to zero for all |z| r, 0 < r < 1. Letting r → 1, we get Thus, f ∈ P τ γ A, B .
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Results Involving Gaussian Hypergeometric Function
Proof.zF a, b; c; z has the series representation given by In view of Theorem 2.2, it suffices to show that S a, b, c, γ :

3.4
From the fact that | a n | ≤ |a| n , we observe that c is real and positive, under the hypothesis

3.6
Using the fact that

3.8
From 1.14 , 3.9 By using the Gauss summation theorem Proof.Let f be of the form 1.1 belong to the class P τ γ A, B .By virtue of Theorem 2.2, it suffices to show that

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Taking into account inequality 2.1 and the relation which is bounded previously by A − B |τ|, in view of inequality 3.12 .
Repeating the previous reasoning for b a, we can improve the assertion of Theorem 3.2 as follows.

B , and if the inequality
In the special case when b 1, Theorem 3.2 immediately yields the following new result.
where k ∈ N.
3.17 Proof.Let f ∈ S. Applying the well-known estimate for the coefficients of the functions f ∈ S, due to de Branges 23 , we need to show that The left-hand side of 3.18 can be written as The second expression of 3.19 , by virtue of the triangle inequality for the pochhammer symbol Now, making use of the relation 3.7 , we get where we are writing n 2 2 n n − 1 5n 4. By repeating the use of 3.7 and the Gauss summation formula, we have

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As a next step, we consider the first expression of equation.By making use of the triangle inequality for the pochhammer symbol as stated in evaluating S 1 , we get

3.23
Now making use of relation 3.7 , we obtain

3.24
where we write n 2 2 n 1 2 2 n 1 1.By repeating the use of 3.7 and the Gauss summation formula, we have

3.25
The proof of Theorem 3.5 now follows by an application of the inequalities of the terms dealing with S 1 , S 2 and inequality 3.17 .
Repeating the previous reasoning for b a, we can improve the assertion of Theorem 3.5 as follows.
Theorem 3.6.Let a, b ∈ C \ {0}.Also, let c be a real number such that c > max{0, 2R a 3 }.If f ∈ S, and if the inequality Proof.By means of 1.17 and 2.13 , the following inequality must be satisfied: Results regarding F a, b; c; z when R c − a − b is positive, zero, or negative are abundant in the literature.In particular when R c − a − b > 0, the function is bounded.This and the zero balanced case R c − a − b 0 are discussed in detail by many authors see 14, 15 .The hypergeometric function F a, b; c; z has been studied extensively by various authors and it plays an important role in Geometric Function Theory.It is useful in unifying various functions by giving appropriate values to the parameters a, b, and c.We refer to 8, 16-19 and references therein for some important results.

1 . 19 When
f z equals the convex function z/ 1 − z , then the operator I a,b,c f in this case becomes zF a, b; c; z .For a 1, b 1 δ, c 2 δ with R δ > −1 then the convolution operator I a,b,c f turns into Bernardi operator

Theorem 3 . 4 .
Let a ∈ C \ {0}.Also, let c be a real number such that c > |a| 1.If f ∈ P τ γ A, B , and if the inequality

Theorem 3 . 5 .
Let a, b ∈ C \ {0}.Also, let c be a real number such that c > |a| |b| 3.If f ∈ S, and if the inequality is satisfied, then I a,b,c f ∈ P τ γ A, B .