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On the Janowski convexity and starlikeness of the confluent hypergeometric function.

1 Introduction

Let A denote the class of analytic functions f defined in the open unit disk D = {z : [absolute value of z] < 1} normalized by the conditions f (0) = 0 = f'(0) - 1. If f and g are analytic in D, then f is subordinate to g, written f (z) < g(z), if there is an analytic self-map w of D satisfying w(0) = 0 and f (z) = g(w(z)). For -1 [less than or equal to] B < A [less than or equal to] 1, let P [A, B] be the class consisting of normalized analytic functions p(z) = 1 + c1z + ... in D satisfying

p(z) < 1 + Az/1 + Bz.

For instance, if 0 [less than or equal to] [beta] < 1, then P [1 - 2[beta], -1] is the class of functions p(z) = 1 + [c.sub.1]z + ... satisfying Re p(z) > [beta] in D.

The class S* [A, B] of Janowski starlike functions [6] consists of f [member of] A satisfying

z f'(z)/f(z) [member of] P[A, B].

For 0 [less than or equal to] [beta] < 1, S*[1 - 2[beta],-1] := S*(b) is the usual class of starlike functions of order [beta]; S*[1 - [beta], 0] := [S*.sub.[beta]] = {f [member of] A : [absolute value of zf'(z)/f (z) - 1] < 1 - [beta]}, and S*[[beta], -[beta]] := S*[[beta]] = {f [member of] A : [absolute value of zf'(z)/f(z) - 1] < [beta][absolute value of zf'(z)/f (z) + 1]}. These classes have been studied, for example, in [2, 3]. A function f [member of] A is said to be close-to-convex of order [beta] [5, 9] if Re (zf'(z)/g(z)) > [beta] for some g [member of] S* := S*(0).

This paper studies the confluent (Kummer) hypergeometric function [PHI](a; c; z) given by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (1.1)

where a, c [member of] C, c [not equal to] 0, -1, -2, ..., and [([lambda]).sub.n] denotes the Pochhammer symbol given by [([lambda]).sub.0] = 1, [([lambda]).sub.n] = [lambda][([lambda] + 1).sub.n-1]. The function [PHI] is a solution of the differential equation

z[PHI]" (a; c; z) + (c - z)[PHI]' (a; c; z) - a[PHI](a; c; z) = 0 (1.2)

introduced by Kummer in 1837 [12]. The Kummer hypergeometric function is an entire analytic function in C and satisfies the relation

c [PHI]' (a; c; z) = a [PHI](a + 1; c + 1; z). (1.3)

When Re c > Re a > 0, [PHI] can be expressed in the integral form

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Several important functions are special cases of the confluent hypergeometric function. These include the Bessel functions, Laguerre polynomials, and the error function. An exposition on O can be found in the book by Temme [12].

Several works [1, 8, 10, 11] have studied geometric properties of the function [PHI](a; c; z), which include studies on its close-to-convexity, starlikeness, and convexity. Miller and Mocanu [8] proved that Re [PHI] (a; c; z) > 0 in D for real a and c satisfying either a > 0 and c [greater than or equal to] a, or a [less than or equal to] 0 and c [greater than or equal to] 1 + ([square root of 1 + [a.sup.2]]). Ponnusamy and Vuorinen [10, Theorem 1.9, p. 77] obtained sufficient conditions for Re [PHI](a; c; z) > [beta], 0 [less than or equal to] [beta] < 1/2. They also determined, as an application of (1.3), conditions that ensure (c/a)([PHI](a; c; z) - 1) is close-to-convex of positive order with respect to the identity function. Additionally they derived conditions for close-to-convexity of z[PHI](a; c;z) with respect to the starlike function z/(1 - z), as well as close-to-convexity of z[PHI](a; c; [z.sup.2]) with respect to the starlike function z/(1 - [z.sup.2]). Constraints on a and c so that [PHI](a; c; z) is convex of positive order are also found in [10, Theorem 5.1, p. 88]. For positive a, Acharya [1, Theorem 4.1.1, p. 50] proved that z[PHI](a; c;z) is starlike and close-to-convex with respect to z and z/(1 - z) if respectively 2c [greater than or equal to] 2a - 1 + ([square root of [a.sup.2] + a + 1]) and 6[c.sup.3] + 3[c.sup.2](a + 6) + 3c(-6[a.sup.2] - 3a + 4) + a(5[a.sup.2] - 221 - 20) [greater than or equal to] 0.

In Section 2 of this paper, sufficient conditions on A, B, a, c are determined that ensure [PHI] satisfies the subordination [PHI](a; c; z) < (1 + Az)/(1 + Bz). Set in this general framework, our findings are obtained through a computationally-intensive methodology with shrewd manipulations. The benefits of such general results are that they not only extend and improved earlier known works, but by judicious choices of the parameters A and B, they give rise to several interesting applications. Examples involving the Laguerre polynomials and the modified Bessel functions are given in Section 3 to illustrate this significance. Sufficient conditions are also obtained for (c/a)[PHI]'(a; c;z) [member of] P[A, B], which readily yields conditions for (c/a)([PHI](a; c; z) - 1) to be close-to-convex. Section 4 gives emphasis to the investigation of [PHI](a; c; z) to be Janowski convex as well as of z[PHI](a; c; z) to be Janowski starlike.

The following lemma is needed in the sequel.

Lemma 1.1. [8, 9] Let [OMEGA] [subset] C, and [PSI] : [C.sup.3] x D [right arrow] C satisfy

[PSI](i[rho], [sigma], [mu] + iv; z) [not member of] [OMEGA]

whenever z [member of] D, [rho] real, [sigma] [less than or equal to] -(1 + [[rho].sup.2])/2 and [sigma] + [mu] [less than or equal to] 0. If p is analytic in D with p(0) = 1, and [PSI](p(z), zp' (z), [z.sup.2]p"(z);z) [member of] [OMEGA] for z [member of] D, then Re p(z) > 0 in D.

In the case [PSI] : [C.sup.2] x D [right arrow] C, then the condition in Lemma 1.1 reduces to [PSI](i[rho], a; z) [not member of] [OMEGA],

[rho] real and [sigma] [less than or equal to] -(1 + [[rho].sup.2])/2.

2 Close-to-convexity of the confluent hypergeometric function

Here is one main result in a general form that has several interesting ramifications.

Theorem 2.1. Let -1 [less than or equal to] B [less than or equal to] 3 - 2 ([square root of 2]) [approximately equal to] 0.171573. Suppose B < A [less than or equal to] 1, and a, c [member of] R satisfy

c - l [greater than or equal to] max {l, [absolute value of 1 +(1 + B)(1 + A)/A - B a]}. (2.1)

Further let A, B, c and a satisfy either the inequality

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.2)

whenever

[absolute value of 2(c - 1)(1 - B)(a(A + B) + A - B) + (1 + B)(A - B + (1 + A)(1 + B)a)] [greater than or equal to] 2(1 - B)[absolute value of A[(1 + a).sup.2] - B[(1 - a).sup.2]], (2.3)

or the inequality

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.4)

whenever

[absolute value of 2(c - 1)(1 - B)(a(A + B) + A - B) + (1 + B)(A - B + (1 + A)(1 + B)a)] < 2(1 - B) (a[(1 + a).sup.2] - B[(1 - a).sup.2]). (2.5)

If (1 + B)[PHI](a; c;z) [not equal to] (1 + A), then [PHI](a; c;z) [member of] P[A, B].

Proof. Define the analytic function p : D [right arrow] C by

p(z) = -(1 - A) - (1 - B)[PHI](z)/(1 + A) - (1 + B)[PHI](z),

where [PHI](z) := 0(a; c; z). Then

[PHI](z) = (1 - A) + (1 + A)p(z)/(1 - B) + (1 + B)p(z), (2.6)

[PHI]'(z) = 2(A - B)p'(z)/((1 - B) + (1 + B)p[(z))'.sup.2] (2.7)

and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (2.8)

Using (2.6)-(2.8), the Kummer differential equation (1.2) yields

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (2.9)

With [OMEGA] = {0}, define [PSI](r, s, t; z) by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (2.10)

It follows from (2.9) that [PSI](p(z), zp' (z), [z.sup.2]p"(z); z) [member of] [OMEGA]. To show Re p(z) > 0 for z [member of] D, from Lemma 1.1, it is sufficient to establish Re [PSI](i[rho], [sigma], [mu] + iv; z) < 0 in D for any real [rho], [sigma] [less than or equal to] - (1 + [[rho].sup.2])/2, and [sigma] + [mu] [less than or equal to] 0.

With z = x + iy [member of] D, it readily follows from (2.10) that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (2.11)

Since [sigma] [less than or equal to] -(1 + [[rho].sup.2])/2, and B [member of] [-1,3 - 2 ([square root of 2])],

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Thus

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

where

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Condition (2.1) shows that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] for [p.sub.1] < 0, it is clear that Q([rho]) < 0 when

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

[absolute value of x], [absolute value of y] < 1. As [y.sup.2] < 1 - [x.sup.2], the above condition holds whenever

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

that is, when

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (2.12)

To establish inequality (2.12), consider the polynomial R given by

R(x) := [mx.sup.2] + nx + r, [absolute value of x] < 1,

where

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

The constraint (2.3) yields [absolute value of n] [greater than or equal to] 2[absolute value of m], and thus R(x) [greater than or equal to] m + r - [absolute value of n]. Now inequality (2.2) readily implies that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Now considers the case of the constraint (2.5), which is equivalent to [absolute value of n] < 2m. Then the minimum of R occurs at x = -n/(2m), and (2.4) yields

R(x) [greater than or equal to] 4mr - [n.sup.2]/4m [greater than or equal to] 0.

Evidently [PSI] satisfies the hypothesis of Lemma 1.1, and thus Re p(z) > 0, that is,

-(1 - A) - (1 - B)[PHI](z)/(1 + A) - (1 + B)[PHI](z) < 1 + z/1 - z.

Hence there exists an analytic self-map w of D with w(0) = 0 such that

- (1 - A) - (1 - B)[PHI](z)/(1 + A) - (1 + B)[PHI](z) = 1 + w(z)/1 - w(z)'

which implies that [PHI](z) < (1 + Az)/(1 + Bz).

Theorem 2.1 gives rise to simple conditions on a and c to ensure [PHI](a; c; z) maps D into a half-plane.

Corollary 2.1. Let a [less than or equal to] 0 and c [greater than or equal to] 2(1 + [a.sup.2]). Then Re [PHI](a; c; z) > a/(a - 1).

Proof. The result follows from Theorem 2.1 by choosing A = -(a + 1)/(a - 1), and B = -1. In this case, condition (2.1) reduces to c [greater than or equal to] 2 which clearly holds for c [greater than or equal to] 2(1 + [a.sup.2]). Also (2.3) and (2.2) respectively reduces to

(1 + [a.sup.2])c - 2(1 + 2[a.sup.2]) [greater than or equal to] 0, (2.13)

and

[(c - 1).sup.2] - 2(c - 1)([a.sup.2] + 1) [greater than or equal to] -2[a.sup.2] - 1. (2.14)

Since c [greater than or equal to] 2 + 2[a.sup.2],

(1 + [a.sup.2])c - 2(1 + 2[a.sup.2]) [greater than or equal to] 2[(1 + [a.sup.2]).sup.2] - 2(1 + 2[a.sup.2]) = 2[a.sup.4] [greater than or equal to] 0,

and

[(c - 1).sup.2] - 2(c - 1)([a.sup.2] + 1) + 2[a.sup.2] + 1 = (c - 2)(c - 2 - 2[a.sup.2]) [greater than or equal to] 0,

which establishes both (2.13) and (2.14).

Remark 2.1. It is noteworthy to compare the result obtained in Corollary 2.1 with a result of Ponnusamy and Vuorinen [10]. With [beta] := a/(a - 1), a [member of] (-1,0], the result of Ponnusamy and Vuorinen [10, Theorem 1.9, p.77] gives Re [PHI](a; c; z) > a/(a - 1) provided c [greater than or equal to] 1 + ([square root of (1 + 3[a.sup.2])/(1 - [a.sup.2])]). Since 1 + ([square root of (1 + 3[a.sup.2])/(1 - [a.sup.2])]) [greater than or equal to] 2 + 2[a.sup.2], it is clear that Corollary 2.1 extends the range of c. Further, since a [less than or equal to] 0, Corollary 2.1 extends the range of [beta] to the whole interval [beta] [member of] [0,1).

Corollary 2.2. Let a > 0 and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Then

Re [PHI](a; c; z) > 1 + [a.sup.2]/(1 + a) [greater than or equal to] 1/2.

Proof. Let A = -[(1 - a).sup.2]/[(1 + a).sup.2], and B = -1 in Theorem 2.1. In this case, (2.1) reduces to c [greater than or equal to] 2, which clearly holds. Now condition (2.3) is trivially true. Thus the result follows from Theorem 2.1 provided (2.2) holds, or equivalently, if

[(c - 1).sup.2] - (c - 1) [absolute value of 1 - [a.sup.2]] [greater than or equal to] [a.sup.2]. (2.15)

Let a [member of] (0,1) and c [greater than or equal to] 2. Then

[(c - 1).sup.2] - (c - 1)[absolute value of 1 - [a.sup.2]] = (c - 1)(c - 2) + (c - 1)[a.sup.2] [greater than or equal to] [a.sup.2].

If a [greater than or equal to] 1, c [greater than or equal to] 1 + [a.sup.2], then

[(c - 1).sup.2] - (c - 1)[absolute value of 1 - [a.sup.2]] = (c - 1)(c - [a.sup.2]) [greater than or equal to] [a.sup.2].

Thus (2.15) holds in either case. From Theorem 2.1, it is evident that [PHI](a; c;z) [member of] P[A, -1], or equivalently,

Re [PHI](a; c; z) > 1 - A/2 = 1 + [a.sup.2]/[(1 + a).sup.2].

Remark 2.2. Corollary 2.2 improves a result in [10, Corollary 1.11, p.77] to the wider range a [member of] (0, [a.sub.0]], where [a.sub.0] = 1.27202 is a root of [a.sub.4] - [a.sub.2] - 1 = 0. It also improves [10, Corollary 1.12, p.78] for a [member of] (0,1).

Corollary 2.3. Let a, c be real such that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Then Re [PHI](a; c; z) > 1/2.

Proof. Put A = 0 and B = -1 in Theorem 2.1. The condition (2.1) reduces to c [greater than or equal to] 2, which holds in all cases. It is sufficient to establish conditions (2.3) and (2.2), or equivalently,

c - 1 - [absolute value of 1 - a] [greater than or equal to] 0, (2.16)

and

[(c - 1).sup.2] - 2(c - 1)[absolute value of 1 - a] - 2a + 1 [greater than or equal to] 0. (2.17)

Consider the case when a [less than or equal to] 0 and c [greater than or equal to] 2 - 2a. Then c - 1 - [absolute value of 1 - a] = c - 2 + a [greater than or equal to] -a [greater than or equal to] 0, and [(c - 1).sup.2] - 2(c - 1)[absolute value of 1 - a] - 2a + 1 = (c - 2)(c - 2 + 2a) [greater than or equal to] 0.

Thus both (2.16) and (2.17) hold.

For a [member of] [0,1] and c [greater than or equal to] 2, then c - 1 - [absolute value of 1 - a] = c - 2 + a [greater than or equal to] a [greater than or equal to] 0, and [(c - 1).sup.2] - 2(c - 1)[absolute value of 1 - a] - 2a + 1 = (c - 2)(c - 2 + 2a) [greater than or equal to] 0.

Finally it is readily established for a [greater than or equal to] 1 and c [greater than or equal to] 2a that c - 1 - [absolute value of 1 - a] = c - a [greater than or equal to] a [greater than or equal to] 0, and [(c - l).sup.2] - 2(c - 1) [absolute value of 1 - a] - 2a + 1 = c(c - 2a) [greater than or equal to] 0.

Remark 2.3. Corollary 2.3 encompasses (for a [greater than or equal to] 1)a result in [10, Theorem 1.13, p.78]. It also expands the range of a to include all real numbers.

Corollary 2.4. Let a [greater than or equal to] -1/2, a [not equal to] 0, and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Then (1 + a)[absolute value of [PHI](a; c; z) - 1] < [absolute value of a]. In particular,

[absolute value of [e.sup.z/2] ((3 - 2z)[I.sub.0] (z/2) + (2z - 1)[I.sub.1] (z/2)) - 3] < 3,

where [I.sub.n] (z) is the modified Bessel function of order n.

Proof. Choose A = [absolute value of a]/(a + 1) and B = 0 in Theorem 2.1. First consider the case when a > 0. The condition (2.1) reduces to c [greater than or equal to] 2a + 3, and condition (2.2), which is equivalent to (c - 1)(c - 2a - 3) [greater than or equal to] 0, clearly holds. Now (2.3) reduces to c [greater than or equal to] 1 + a, which holds because c [greater than or equal to] 2a + 3, a > 0. From Theorem 2.1 it is evident that [PHI](a; c; z) < 1 + az/(1 + a), that is, [absolute value of [PHI](a; c; z) - 1] < a/(1 + a).

Now consider the case when a [member of] [-1/2,0). Then (2.1) gives c [greater than or equal to] 2. Conditions (2.2) and (2.3) respectively become

c(c - 1) - 2[absolute value of (c - 1)(1 + a)] > 0, (2.18)

and

[absolute value of - 2a(c - 1)] + 2a(1 + a) > 0. (2.19)

Since c [greater than or equal to] 2 and a [member of] [-1/2,0),

c(c - 1) - 2[absolute value of (c - 1)(1 + a)] = c(c - 1) - 2(c - 1)(1 + a) = (c - 1)(c - 2 - 2a) > 0, and

[absolute value of - 2a(c - 1)] + 2a(1 + a) = -2a(c - 1) + 2a + 2[a.sup.2] = -2a(c - 2) + 2[a.sup.2] > 0, which established both (2.18) and (2.19). Again from Theorem 2.1 it follows that

(1 + a)[absolute value of [PHI](a; c; z) - 1] < -a.

An application of the identity

[PHI](-1/2;2;z) = [e.sup.z/2]/3 ((3 - 2z) [I.sub.0](z/2) + (2z - 1)[I.sub.1](z/2))

establishes the final assertion.

Theorem 2.2. Let 3 - 2 ([square root of 2]) < B < A [less than or equal to] 1, and a, c [member of] R satisfy

c - l [greater than or equal to] max{1, [absolute value of 1 + (1 + B)(1 + A)/A - B a]}. (2,20)

Suppose A, B, c and a satisfy

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.21)

whenever

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (2.22)

or the inequality

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.23)

when

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (2.24)

If (1 + B)[PHI](a; c; z) [not equal to] (1 + A), then [PHI](a; c; z) [member of] [A,B].

Proof. Proceeding similarly as in the proof of Theorem 2.1, consider Re [PSI](i[rho], [sigma], [mu] + iv;z) as given in (2.11). For [sigma] [less than or equal to] - (1 + [[rho].sup.2])/2, [rho] [member of] R, and B [greater than or equal to] 3 - 2[square root of 2],

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

With z = x + iy [member of] D, and [mu] + [sigma] < 0, it follows that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

where

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

As per the proof of Theorem 2.1, observe that the constraint (2.20) implies that [p.sub.2] < 0. Thus [Q.sub.1]([rho]) < 0 for all [rho] [member of] R provided [q.sup.2.sub.2] [less than or equal to] 4[p.sub.2][r.sub.2], that is,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

[absolute value of x], [absolute value of y] < 1. With [y.sup.2] < 1 - [x.sup.2], it is sufficient to show

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

for [absolute value of x] < 1. The above inequality is equivalent to showing

[R.sub.1] (x) := [m.sub.1] [x.sup.2] + [n.sub.1]x + [r.sub.1] [greater than or equal to] 0, (2.25)

where

[m.sub.1] := 1 + 2a A + B/A - B + [a.sup.2],

[n.sub.1] := -2((c-1)(a(a + B)/(A - B) + 1) + 8B(1 - B)/[(1 + B).sup.3](1 + (1 + A)(1 + B)/A - B a)),

[r.sub.1] := [(c - 1).sup.2] + (c - 1) 16B(1 - B)/[(1 + B).sup.3] - [a.sup.2][(1 - AB).sup.2]/[(A - B).sup.2].

If (2.22) holds, then [absolute value of [n.sub.1]] [greater than or equal to] 2[absolute value of [m.sub.1]]. Since [R.sub.1] is increasing, then [R.sub.1](x) [greater than or equal to] [m.sub.1] + [r.sub.1] - [absolute value of n1], which is nonnegative from (2.21). On the other hand, if (2.24) holds, then [absolute value of [n.sub.1]] < 2[absolute value of [m.sub.1]], R1 (x) [greater than or equal to] (4[m.sub.1][r.sub.1] - [n.sup.2.sub.1])/4[m.sub.1], and (2.23) implies [R.sub.1](x) [greater than or equal to] 0. Either case establishes (2.25).

The following results are immediate consequences from relation (1.3) and respectively Theorem 2.1 and Theorem 2.2.

Theorem 2.3. Let -1 [less than or equal to] B [less than or equal to] 3 - 2([square root of 2]) [approximately equal to] 0.171573. Suppose B < A [less than or equal to] 1, a, c [member of] R

with a [not equal to] 0 and satisfying

c [greater than or equal to] max{1, [absolute value of 1 + (1 + B)(1 + A)/A - B (a + 1)]}.

Further let A, B, c and a satisfy either

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

whenever

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

or the inequality

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

when

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

If (1 + B)[PHI](a + 1; c + 1; z) [not equal to] (1 + A), then (c/a)[PHI]' (a; c; z) [member of] P[A, B],

Theorem 2.4. Let 3 - 2 ([square root of 2]) < B < A [less than or equal to] 1. Suppose a, c [member of] R, a [not equal to] 0, such that

c [greater than or equal to] max{1, [absolute value of 1 + (1 + B)(1 + A)/A - B (1 + a)]}.

Further let A, B, c and a satisfy

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

when

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

or the inequality

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

when

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

If (1 + B)[PHI](a + 1; c + 1; z) [not equal to] (1 + A), then (c/a) [PHI]'(a; c; z) [member of] P[A, B],

With B = -1, the following results are easily deduced from Theorem 2.3 by choosing respectively A = -(a + 2)/a, A = -[a.sup.2]/[(2 + a).sup.2] and A = 0.

Corollary 2.5. Let a [less than or equal to] -1, and c [greater than or equal to] 1 + 2[(1 + a).sup.2]. Then (c/a)( [PHI](a; c; z) - 1) is close-to-convex of order (a + 1)/a with respect to the identity function.

Corollary 2.6. Let a > -1, a [not equal to] 0, and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Then (c/a)([PHI](a; c; z) - 1) is close-to-convex of order (2 + 2a + [a.sup.2])/[(2 + a).sup.2] with respect to the identity function.

Corollary 2.7. Let a be a nonzero real number, and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Then Re(c/a)[PHI]'(a; c; z) > 1/2.

3 Examples

The associated Laguerre polynomial [L.sup.[alpha].sub.n] [4] is given by the series

[L.sup.[alpha].sub.n](z) := [n.summation over (k=0)] [(-1).sup.k] [TAU](n + [alpha] + 1)/(n - k)![TAU]([alpha] + k + 1)k! [z.sup.k].

It relates to the confluent hypergeometric function via the identity

[([alpha] + 1).sub.n][PHI](-n; [alpha] + 1; z) = n! [L.sup.[alpha].sub.n] (z).

Thus

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

The associated Laguerre polynomial [L.sup.[alpha].sub.n] is used to illustrate the significance of Theorem 2.1.

Example 3.1. For every m = 2,3, ...,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (3.1)

provided [(1 + m).sup.2](2 - 2m + [m.sup.2])[(1 + [m.sup.4]/4).sub.m] [not equal to] m! [(1 - m).sup.2](2 + 2m + [m.sup.2]) [L.sup.m4/4.sub.m] (z).

To verify the subordination given in (3.1), consider p : D [right arrow] C given by

p(z) := -(1 - 1/[(1 - m).sub.2]) - (1 - 1/[(1 + m).sup.2]) [[PHI].sub.1](z)/(1 + 1/[(1 - m).sub.2]) - (1 + 1/[(1 + m).sup.2]) [[PHI].sub.1](z),

where [[PHI].sub.1] (z) := [PHI] (-m; 1 + [m.sup.4]/4; z). Proceeding as in the proof of the Theorem 2.1 with A = 1/[(1 - m).sup.2] and B = 1/[(1 + m).sup.2], then (2.11) reduces to

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (3.2)

Now [sigma] [less than or equal to] -(1 + [[rho].sup.2])/2 yields

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Taking logarithmic derivatives, it is evident that h is increasing for [rho] [greater than or equal to] 0, and thus h([rho]) [greater than or equal to] ([m.sup.2] + 2m + 2)/2m(m + 2). Now as [mu] + [sigma] < 0, (3.2) yields

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

where

[p.sub.1] = - [m.sup.4]/8 (1 + x) < 0,

[q.sub.1] = -1/4 [m.sup.2] ([m.sup.2] - 2) y,

[r.sub.1] = -[m.sup.2] + 2m + 2/2m(m + 2) - [m.sup.4]/8 + 1/8 [([m.sup.2] - 2).sup.2] x.

As [absolute value of x] < 1, [absolute value of y] < 1 and [y.sup.2] < 1 - [x.sup.2], it follows that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Since [p.sub.1] < 0, the above inequality implies that max([p.sub.1][[rho].sup.2] + [q.sub.1][rho] + [r.sub.1]) = (4[p.sub.1][r.sub.1] - [q.sup.2.sub.1])/(4[p.sub.1]) < 0 over [rho] [member of] R. Hence Re [PSI](i[rho], [sigma], [mu] + iv; z) < 0. The final assertion now follows readily by using the same arguments as in the proof of Theorem 2.1.

The next example illustrates both Theorem 2.1 and Theorem 2.2 in the sense that the real parameters satisfy the conditions of each theorem within different constraint intervals.

Example 3.2. If a [member of] (-[infinity], -1) and c [greater than or equal to] max{2,1 + [a.sup.4]/4}, then

[PHI](a; c; z) < 1 + z/[(1 + a).sup.2]/1 + z/[(1 - a).sup.2], z [member of] D, (3.3)

provided [(1 - a).sup.2] (2 + 2a + [a.sup.2]) [not equal to] [(1 + a).sup.2] ([a.sup.2] - 2a)[PHI](a; c; z).

To establish the subordination (3.3), define the analytic function p : D [right arrow] C by

p(z) := -(1 - 1/[(1 + a).sup.2]) - (1 - 1/[(1 - a).sup.2])[PHI](a; c; z)/(1 + 1/[(1 + a).sup.2]) - (1 + 1/[(1 - a).sup.2])[PHI](a; c; z).

Proceeding similarly as in the proof of the Theorem 2.1 with A = 1/[(1 + a).sup.2] and B = 1/[(1 - a).sup.2], the equivalent form for (2.11) is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3.4)

Now [sigma] [less than or equal to] -(1 + [[rho].sup.2])/2 yields

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

where h([rho]) = [(1 + [[rho].sup.2]).sup.2]/([(a - 2).sup.2][a.sup.2] + [((a - 2)a + 2).sup.2][[rho].sup.2]) > 0. By taking logarithmic derivatives, it follows that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (3.5)

We next consider two cases. First is when a [member of] (-[infinity], - ([square root of 2])], that is, B [less than or equal to] 3 - 2 ([square root of 2]).

In this case, it follows from (3.5) that h is increasing for [rho] [greater than or equal to] 0, and hence h([rho]) [greater than or equal to] l/([(a - 2).sup.2][a.sup.2]). Thus

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

where

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Since [absolute value of x] < 1, [absolute value of y] < 1 and [y.sup.2] < l - [x.sup.2], a computation gives

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Since max([p.sub.1][[rho].sup.2] + [q.sub.1]p + [r.sub.1]) = (4[p.sub.1][r.sub.1] - [q.sup.2.sub.1])/(4[p.sub.1]) < 0 for [rho] [member of] R, it follows that Re [PSI](i[rho], [sigma], [mu] + iv;z) < 0. The final assertion for the case a [member of] (-[infinity], - ([square root of 2])] follows by adopting the same arguments used in the proof of Theorem 2.1.

Next consider the case when a [member of] (- ([square root of 2]), -1), that is, the case B [greater than or equal to] 3 - 2 ([square root of 2]). For this range of a, the expression [a.sup.4] - 4[a.sup.3] + 8a - 4 is negative, and hence h' given in (3.5) is non-negative when [rho] > 0 and [[rho].sup.2] [greater than or equal to] -([a.sup.4] - 4[a.sup.3] + 8a - 4)/[([a.sup.2] - 2a + 2).sup.2]. Together these imply that

h([rho]) [greater than or equal to] 16[(a - 1).sup.2]/[([a.sup.2] - 2a + 2).sup.4].

It is immediate from (3.4) that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

where

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

We need to show that [q.sup.2.sub.2] - 4[p.sub.2][r.sub.2] < 0. Since [absolute value of x] < 1, [absolute value of y] < 1 and [y.sup.2] < 1 - [x.sup.2], a computation yields

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Evidently, the conclusion follows if the expression

[M.sub.a] = [([a.sup.2] - 2a + 2).sup.3] ([a.sup.4] - 2[a.sup.2] + 2) + 32a(a - 2)[(a - 1).sup.2]

is positive, which clearly holds for a [member of] (-([square root of 2]), -1).

4 Janowski starlikeness of the confluent hypergeometric function

This section looks at finding conditions to ensure a normalized confluent hypergeometric function z[PHI](a; c; z) is Janowski starlike. For this purpose, we first derive sufficient conditions for 0(a; c; z) to be Janowski convex, after which an application of relation (1.3) yields conditions for z[PHI](a; c ; z) [member of] S* [A, B].

Theorem 4.1. Let a, c [member of] R be such that (A - B)[PHI]'(a; c, z) [not equal to] (1 + B)z[PHI]"(a; c; z), -1 [less than or equal to] B < A [less than or equal to] 1. Suppose

c(1 + B) [greater than or equal to] (1 + B)[absolute value of 1 + (1 + a)(1 + B)/A - B] - (A - B + 1). (4.1)

Further let A, B, c and a satisfy either

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (4.2)

whenever

[absolute value of 1 - B(A - B) + (2B - (A - B))(1 + [B.sup.2])) 1 + a/A - B + c(1 - [B.sup.2]) (1 + B(1 + a)/A - B)] [less than or equal to] 1, (4.3)

or the inequality

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (4.4)

when

[absolute value of 1 - B(A - B) + (2B - (21 - B)(l + [B.sup.2])) 1 + a/A - B + c(1 - [B.sup.2])(1 + B(1 +a)/A - B)] [greater than or equal to] 1 (4.5)

If 0 [not member of] [PHI]' (D),0 [not member of] [PHI]" (D), then

1 + z[PHI]"(a; c, z)/[PHI]'(a; c, z) < 1 + Az/1 + Bz.

Proof. For convenience, write [PHI](z) := [PHI](a; c; z), and define an analytic function p : D [right arrow] C by

p(z) := (A - B)[PHI]'(z) + (1 + B)z[PHI]"(z)/(A - B)[PHI]'(z) - (1 + B)z[PHI]"(z).

Then

z[PHI]"(z)/[PHI]'(z) = (A - B)(p(z) - 1)/(p(z) + 1) + B(p(z) - 1)' (4.6)

and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Thus

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

and together with (4.6) imply that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (4.7)

A differentiation of (1.2) leads to

z[PHI]"'(z) + (c - z + 1)[PHI]"(z) - (1 + a)[PHI]'(z) = 0.

It follows that

(z[PHI]"(z)/[PHI]'(z))(z[PHI]"(z)/[PHI]'(z)) + (c - z + 1)z[PHI]"(z)/[PHI]'(z) - (1 + a)z = 0 (4.8)

Using (4.6) and (4.7), (4.8) yields

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

that is,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (4.9)

Denote by [PSI](p(z), zp' (z), z) := zp'(z) + [F.sub.1] [(p(z)).sup.2] + [F.sub.2] p(z) + [F.sub.3], where

[F.sub.1] = (A - B)/2 + (c - z)(1 + B)/2 - (1 + a)z[(1 + B).sup.2]/2(A - B),

[F.sub.2] = -(A - B) - (c - z)B - (1 + a)(1 - [B.sup.2])/(A - B) z,

[F.sub.3] = (A - B)/2 + (c - z)(-1 + B)/2 - (1 + a)z[(1 - B).sup.2]/2(A - B).

With [OMEGA] = {0}, (4.9) yields [PSI](p(z), zp' (z), z) [member of] [OMEGA]. Now with z = x + iy [member of] D, let

[G.sub.1] := Re([f.sub.1]) = (A - B/2 + (c - x)(1 + B)/2 - (1 + a)x[(1 + B).sup.2]/2(A - B)).

[G.sub.2] := Re(i[F.sub.2]) = -yB + (1 + a)(1 -[B.sup.2])/(A - B)y,

[G.sub.3] := Re([F.sub.3]) = (A - B/2 +(c - x)(-1 + B)/2 -(1 + a)x[(1 - B).sup.2]/2(A - B)) =1/2(A - B - c(1 - B) + x(1 - B)(1 - (1 + a)(1 - B)/A - B)).

For [sigma] [less than or equal to] -(1 + [[rho].sup.2])/2, [rho] [member of] R,

Re [PSI](i[rho], [sigma], z) = [sigma] - [G.sub.1][[rho].sup.2] + [G.sub.2][rho] + [G.sub.3] [less than or equal to] - 1/2((1 + 2[G.sub.1]) [[rho].sup.2] - 2[G.sub.2][rho] - 2[G.sub.3] + 1):= Q([rho]).

Note that condition (4.1) implies (1 + 2[G.sub.1])/2 > 0. In this case, Q has a maximum at [rho] = [G.sub.2]/(1 + 2[G.sub.1]). Thus Q([rho]) < 0 for all real [rho] provided

[G.sup.2.sub.2] [less than or equal to] (1 + 2[G.sub.1])(1 - 2[G.sub.3]), [absolute value of x], [absolute value of y] < 1.

Since [y.sup.2] < 1 - [x.sup.2], it is left to show that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

[absolute value of x] < 1. The above inequality is equivalent to

H(x) := [x.sup.2] - 2[h.sub.2] (A, B)x + [h.sub.e] (A, B) [greater than or equal to] 0, (4.10)

where

[h.sub.2](A,B) = l - BA + [B.sup.2] + (2B -(A - B)(1 + [B.sup.2]))(1 + a)/A - B + (1 + B(1 + a)/A - B)(1 - [B.sup.2])c,

[h.sub.3](A,B) = (1 + A - B + c(1 + B))(1 - A + B + c(1 - B)) - [((1 + a)(1 - [B.sup.2])/A - B -B).sup.2].

To establish (4.10), first consider the case when (4.3) holds. Then [absolute value of [h.sub.2]] [less than or equal to] 1, and hence H'(x) = 0 at x = [h.sub.2] [member of] (-1,1). Since H"(x) > 0, H has a minimum at x = [h.sub.2] and from (4.2), it follows that

H(x) [greater than or equal to] H([h.sub.2]) = [h.sup.2.sub.2] - 2[h.sup.2.sub.2] + [h.sub.3] = [-.sup.h2.sub.2] + [h.sub.3] [greater than or equal to] 0.

Now consider the case when (4.5) holds. In this case [absolute value of [h.sub.2]] [greater than or equal to] 1, and H'(x) = 2(x - [h.sub.2]) [less than or equal to] 2(1 - [h.sub.2]) [less than or equal to] 0. Hence H is decreasing. From (4.4), it is evident that

H(x) [greater than or equal to] H(1) = 1 - 2[h.sub.2] + [h.sub.3] [greater than or equal to] 1 - 2[absolute value of [h.sub.2]] + [h.sub.3] [greater than or equal to] 0.

Thus [PSI] satisfies the hypothesis of Lemma 1.1, and hence Re p(z) > 0, or equivalently

(A - B)[PHI]' + (1 - B)z[PHI]"/(A - B)[PHI]' - (1 + B)z[PHI]" < 1 + z/1 - z.

By definition of subordination, there exists an analytic self-map w of D with w(0) = 0 and

(A - B)[PHI]'(z) + (1 - B)z[PHI]" (z)/(A - B)[PHI]'(z) - (1 + B)z[PHI]"(z) = 1 + w(z)/1 - w(z)

A simple computation shows that

1 + z[PHI]"(z)/[PHI]'(z) = 1 + Aw(z0/1 + Bw(z),

and hence

1 + z[PHI]"(z)/[PHI]'(z) < 1 + Az/1 + Bz.

The relation (1.3) also shows that

z(z[PHI](a; c; z))'/z[PHI](a; c; z) = 1 + z[PHI]"(a - 1; c - 1; z)/[PHI]'(a - 1; c - 1; z).

Together with Theorem 4.1, it immediately yields the following result for z[PHI](a; c; z) [member of] S* [A, B].

Theorem 4.2. Let a and c be real numbers such that (A - B)[PHI]'(a - 1; c - 1, z) [not equal to] (1 + B)z[PHI]" (a - 1; c - 1;z), -1 [less than or equal to] B < A [less than or equal to] 1. Suppose

c(1 + B) [greater than or equal to] (1 + B) [absolute value of 1 + a(1 + B)/A - B] - (A - 2B). (4.11)

Further let A, B, c and a satisfy either

(l - B(A - B) + (2B - (A - B)(l + [B.sup.2])) a/A - B + (c - 1)[(1 - [B.sup.2])(1 + Ba/A - B)).sup.2] [less than or equal to] (A - 2B + c(1 + B))(2B - A + c(l - B)) - [(a(1 - [B.sup.2])/A - B - B).sup.2] (4.12)

whenever

[absolute value of 1 - B(A - B) + (2B - (A - B)(1 + [B.sup.2])) a/A - B + (c - 1)(1 - [B.sup.2])(1 + Ba/A - B)] [less than or equal to] 1, (4.13)

or

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (4.14)

when

[absolute value of 1 - B(A - B) + (2B - (A - B)(l + B2)) a/A - B + (1 + Ba/A - B)(1 - [B.sup.2])(c - 1)] [greater than or equal to] 1. (4.15)

Then z[PHI](a; c; z) [member of] S*[A, B].

Remark 4.1. Choosing A = 1 - 2[beta], [beta] [member of] [0,1), and B = -1, then Theorem 4.1 gives [10, Theorem 5.1, p. 88], while Theorem 4.2 gives [10, Theorem 5.3, p. 88]. For a = 1, c = 1 + [delta], [delta] [greater than or equal to] 1, Theorem 4.1 gives sufficient conditions for

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

to be Janowski convex. This is a generalization of a result in [10, Corollary 5.8, p. 89].

Writing f (z) = z[PHI](a; c; z), and g(z) = f ([z.sup.2])/z, it follows that

zg'(z)/g(z) = 2 [z.sup.2]f'([z.sup.2])/f([z.sup.2]) - 1. (4.16)

Now (4.16) and Theorem 4.2 immediately yield the following result.

Corollary 4.1. Let a and c be real numbers such that (A - B)[PHI]'(a + 1; c + 1; z) [not equal to] (1 + B)z[PHI]" (a + 1; c + 1; z) and (4.11) holds for -1 [less than or equal to] B < 2 A - B [less than or equal to] 1. Further let A, B, c and a satisfy either (4.12) when (4.13) holds, or (4.14) when (4.15) holds. Then z[PHI](a; c; [z.sup.2]) [member of] S* [2A - B,B].

Remark 4.2. Particular choice of A and B yield the following ramifications.

(i) For A = 1 - 2[beta] and B = -1, Corollary 4.1 reduces to [10, Corollary 5.13, p. 93].

(ii) For a = 1/2, Corollary 4.1 is useful in obtaining conditions for the generalized error function z[PHI] (1/2; c, z) [member of] S* [2A - B, B].

Acknowledgment. The work presented here was supported by a FRGS and RU research grants from Universiti Sains Malaysia. The authors are thankful to the referee for the insightful comments that helped improve the clarity of this manuscript.

References

[1] A. P. Acharya, Univalence criteria for analytic functions and applications to hypergeometric functions, Ph.D diss., University of WUrzburg, 1997.

[2] R. M. Ali, V. Ravichandran and N. Seenivasagan, Sufficient conditions for Janowski starlikeness, Int. J. Math. Math. Sci. 2007, Art. ID 62925, 7 pp.

[3] R. M. Ali, R. Chandrashekar and V. Ravichandran, Janowski starlikeness for a class of analytic functions, Appl. Math. Lett. 24 (2011), no. 4, 501-505.

[4] W. W. Bell, Special functions for scientists and engineers, D. Van Nostrand Co., Ltd., London, 1968.

[5] A. W. Goodman, Univalent functions. Vol. I & II, Mariner, Tampa, FL, 1983.

[6] W. Janowski, Some extremal problems for certain families of analytic functions. I, Ann. Polon. Math. 28 (1973), 297-326.

[7] S. S. Miller and P. T. Mocanu, Univalence of Gaussian and confluent hypergeometric functions, Proc. Amer. Math. Soc. 110 (1990), no. 2, 333-342.

[8] S. S. Miller and P. T. Mocanu, Differential subordinations and inequalities in the complex plane, J. Differential Equations 67 (1987), no. 2,199-211.

[9] S. S. Miller and P. T. Mocanu, Differential subordinations, Monographs and Textbooks in Pure and Applied Mathematics, 225, Dekker, New York, 2000.

[10] S. Ponnusamy and M. Vuorinen, Univalence and convexity properties for confluent hypergeometric functions, Complex Variables Theory Appl. 36 (1998), no. 1, 73-97.

[11] St. Ruscheweyh and V. Singh, On the order of starlikeness of hypergeometric functions, J. Math. Anal. Appl. 113 (1986), no. 1,1-11.

[12] N. M. Temme, Special functions, A Wiley-Interscience Publication, Wiley, New York, 1996.

School of Mathematical Sciences, Universiti Sains Malaysia, 11800 USM Penang Malaysia

email:rosihan@usm.my

Department of Mathematics, King Faisal University, Ahsaa 31982, Saudi Arabia

email:smondal@kfu.edu.sa

Department of Mathematics, University of Delhi, Delhi 110007, India

email:vravi@maths.du.ac.in

Received by the editors in May 2013--In revised form in September 2014.

Communicated by H. De Schepper.

2010 Mathematics Subject Classification : 30C45,30C80,40G05.
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