# Inclusion of the Generalized Bessel Functions in the Janowski Class.

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 = g [omicron] u >. 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 + [c.sub.1]z + ... in D satisfying

P(Z) < 1 + Az/1 + Bz. (1)

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.sup.*][A, B] of Janowski starlike functions [1] consists of f [member of] A satisfying

Zf'(z)/f(z) [member of] P [A,B]. (2)

For 0 [less than or equal to] [beta] < 1, [S.sup.*][1-2[beta], -1] := [S.sup.*]([beta]) is the usual class of starlike functions of order [beta]; [S.sup.*][1 - [beta], 0] = [S.sup.*.sub.[beta]] = {f [member of] A : [absolute value of (zf'(z)/f(z) - 1)] < 1 - [beta]}; and [S.sup.*] [[beta], -[beta]] := [S.sup.*] [[beta]] = {f [member of] A : [absolute value of (zf'(z)/f(z)-1)] < [beta][absolute value of (z/'(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] [4,5] if Re (zf' (z)/g(z)) > [beta] for some g [member of] [S.sup.*] := [S.sup.*(0).]

This article studies the generalized Bessel function [u.sub.p](z) = [u.sub.p,b,c](z) given by the power series

[U.sub.p](z) = [sub.0][F.sub.1]([kappa], -c/4z) = [[infinity].summation over (k=0)] [(-1).sup.k][c.sup.k]/[4.sup.k][([kappa]).sub.k] [z.sup.k]/k!, (3)

where [kappa] = p + (b+1)/2 [not equal to] 0, -1, -2, -3, ... The function [u.sub.p](z) is analytic in D and solution of the differential equation

4[z.sup.2]u" (z) + 4[kappa]zu' (z) + czu (z) = 0, (4)

if b, p, c in C, such that [kappa] = p + (b + 1)/2 [not equal to] 0,-1,-2,-3, ..., and z [member of] D. This normalized and generalized Bessel function of the first kind of order p also satisfies the following recurrence relation:

4[kappa][u'.sub.p] (z) = -[cu.sub.p+1](z), (5)

which is useful tool to study several geometric properties of Up. There have been several works [6-11] studying geometric properties of the function [u.sub.p](z), for example, on its close-to convexity, star likeness, and convexity, and radius of star likeness and convexity.

In Section 2 of this paper, sufficient conditions on A, B, c, and [kappa] are determined that will ensure up satisfies the subordination [u.sub.p](z) < (1+Az)/(1+Bz). It is to be understood that a computationally intensive methodology with shrewd manipulations is required to obtain the results in this general framework. The benefits of such general results are that, by judicious choices of the parameters A and B, they give rise to several interesting applications, which include extending the results of previous works. Using this subordination result, sufficient conditions are obtained for (-4[kappa]/c)u (z) [member of] P[A, B], which next readily gives conditions for (-4[kappa]/c)([u.sub.p](z) - 1) to be close-to-convex. Section 3 gives emphasis to the investigation of [u.sub.p](z) to be Janowski convex as well as of z[u.sub.p](z) to be Janowski starlike. The following lemma is needed in sequel.

Lemma 1 (see [5,12]). Let [OMEGA] [subset] C and [PHI] : [C.sup.2] x D [right arrow] C satisfy

[PHI](i[rho],[alpha];z) [not member of] [OMEGA] (6)

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

In the case [PHI] : [C.sup.3] x D [right arrow] C, then the condition in Lemma 1 is generalized to

[PHI](ip, [sigma], [mu] + iv; z) [not member of] [OMEGA]. (7)

[rho] is real, [sigma] + [mu] [less than or equal to] 0, and [sigma] [less than or equal to] -(1 + [[rho].sup.2])/2.

2. Close-To-Convexity of Generalized Bessel Functions

In this section, one main result on the close-to-convexity of the generalized Bessel function with several consequences is discussed in detail.

Theorem 2. 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 c,[kappa] [member of] R satisfy

k - 1 [greater than or equal to] (1+B)(1+A)/4(A - B)[absolute value of (c)]. (8)

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

[mathematical expression not reproducible] (9)

whenever

[absolute value of (2 (k - 1) (1 - B) (A + B) c + [(1 + B).sup.2] (1 + A) c)] [greater than or equal to] 1/2(a-b)(1-b)[c.sup.2] (10)

or the inequality

[mathematical expression not reproducible] (11)

whenever

[absolute value of (2 ([kappa] - 1) (1 - B) (A + B) c + [(1 + B).sup.2] (1 + A) c)] < 1/2(A - B)(1 - B)[c.sup.2]. (12)

If (1+B)[u.sub.p](z) [not equal to] (1 + A) for all z [member of] D, then up(z) [member of] P[A,B].

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

P(Z) = (1-A)-(1-B) [u.sub.p](z)/(1 + A)-(1+B)[u.sub.p](z). (13)

Then, a computation yields

[mathematical expression not reproducible]. (14)

Thus, using identities (14), the Bessel differential equation (4) can be rewritten as

[mathematical expression not reproducible]. (15)

Assume [OMEGA] = {0}, and define [PHI](r, s, t; z) by [PHI] (r, s, t; z)

:= 2(1+B)/(1 - B) + (1 + B)r [S.sup.2] + [kappa]S ((1-B) + (1 + B)r)((1 - A) + (1 + A)r)/8(A - B) cz. (16)

It follows from (15) that [PHI](p(z),zp'(z),[z.sup.2]p"(z);z) [member of] [OMEGA]. To ensure Re p(z) > 0 for z [member of] D, from Lemma 1, it is enough to establish Re [PHI](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 in (16), a computation yields

Re [PHI] (i[rho], [mu] + iv; z)

[mathematical expression not reproducible]. (17)

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]. (18)

Thus

Re [PHI](i[rho], [sigma], [mu] + iv; z)

[mathematical expression not reproducible], (19)

where

[mathematical expression not reproducible]. (20)

Condition (8) shows that

[mathematical expression not reproducible]. (21)

Since [max.sub.p[member of]R] {[p.sub.1][p.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]) for [p.sub.1] < 0, it is clear that Q([rho]) < 0 when

[mathematical expression not reproducible], (22)

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

[mathematical expression not reproducible], (23)

that is, when

[mathematical expression not reproducible]. (24)

To establish inequality (24), consider the polynomial R given by

R (x) := m[x.sup.2] + nx + r, [absolute value of (x)] < 1, (25)

where

[mathematical expression not reproducible] (26)

Constraint (10) 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 (9) readily implies that

[mathematical expression not reproducible]. (27)

Now consider the case of constraint (12), which is equivalent to [absolute value of (n)] < 2m. Then the minimum of R occurs at x = -n/(2m), and (11) yields

R(x) [greater than or equal to] 4mr - [n.sup.2]/4m [greater than or equal to] 0. (28)

Evidently [PHI] satisfies the hypothesis of Lemma 1, and thus Re p(z) > 0; that is,

(1-A)-(1-B)[u.sub.p] (z)/(1+A)-(1 + B)up (z) < 1-z/1 -z. (29)

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

(1-A)-(1-B)[u.sub.p] (z)/(1 + A)-(1 + B)[u.sub.p] (z) = 1+w(z)/1-w(z), (30)

which implies that [u.sub.p](z) < (1 + Az)/(1 + Bz).

Theorem 2 gives rise to simple conditions on c and [kappa] to ensure [u.sub.p](z) maps D into a half plane.

Corollary 3. Let c [less than or equal to] 0 and [kappa] [greater than or equal to] max{5/4,1 + [c.sup.2]/2}. Then Re [u.sub.p](z) > c/(c - 1).

Proof. Choose A = -(c + 1)/(c - 1) and B = - 1 in Theorem 2. Then condition (8) is equivalent to [kappa] [greater than or equal to] 1 and (10) reduces to [kappa] [greater than or equal to] 5/4, and clearly both hold for [kappa] [greater than or equal to] max{5/4,1 + [c.sup.2]/2}. The proof will complete if hypothesis (9) holds; that is,

[([kappa]-1).sup.2] [greater than or equal to] 1/2([kappa]-1)[c.sup.2]. (31)

Since [kappa] [greater than or equal to] 1 + [c.sup.2]/2, it follows that

[([kappa]-1).sup.2] - 1/2([kappa] - 1)[c.sup.2] = ([kappa] - 1) ([kappa] - 1 - [c.sup.2]/2) [greater than or equal to] 0, (32)

which establishes (31).

Corollary 4. Let c [member of] R. Then if [kappa] [greater than or equal to] [absolute value of (c)]/2, then Re [u.sub.p](z) > 1/2.

Proof. Put A = 0 and B = -1 in Theorem 2. Condition (8) reduces to [kappa] [greater than or equal to] 1, which holds in all cases. It is sufficient to establish conditions (10) and (9) or, equivalently,

4([kappa] - 1) - [absolute value of (c)] [greater than or equal to] 0, [([kappa] -1).sup.2] - 1/2 ([kappa]-1)[absolute value of (c)] [greater than or equal to] 0. (33)

The hypothesis [kappa] - 1 [greater than or equal to] [absolute value of (c)]/2 implies that 4([kappa]-1)-[absolute value of (c)] [greater than or equal to] [absolute value of (c)] [greater than or equal to] 0 and [([kappa] -1).sup.2] - (1/2)([kappa] - 1)[absolute value of (c)] [greater than or equal to] ([kappa] - 1)([kappa] - 1 - [absolute value of (c)]/2) [less than or equal to] 0.

Next theorem gives the sufficient condition for close-toconvexity when B [greater than or equal to] 3 - [square root of 2]

Theorem 5. Let 3-2[square root of 2] [less than or equal to] B < A [less than or equal to] 1 and c,k [member of] R satisfys

[kappa] - 1 [greater than or equal to] (1 + B)(1 + A)/4(A - B) [absolute value of (C)]. (34)

Suppose A, B, [kappa], and c satisfy either the inequality

[mathematical expression not reproducible] (35)

whenever

[mathematical expression not reproducible] (36)

or the inequality

[mathematical expression not reproducible] (37)

whenever

[absolute value of (([kappa] -1) (A + B) [(1 + B).sup.3] + 8B(1 - [B.sup.2]) (1 + A)) c)] < [c.sup.2]/4 (A - B) [(1 + B).sup.3]. (38)

If (1 + B)[u.sub.p](z) [not equal to] (1 + A) for all z [member of] D, then [u.sub.p](z) [member of] P[A,B].

Proof. First, proceed similarly to the proof of Theorem 2 and derive the expression of Re [PHI](i[rho], [sigma], [mu] + iv;z) as given in (17). Now, 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], (39)

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

[mathematical expression not reproducible], (40)

where

[mathematical expression not reproducible], (41)

Observe that inequality (34) implies that [p.sub.2] < 0. Thus [Q.sub.1]([rho]) < 0 for all [rho] [member of] R provided [q.sup.2.sub.2] < 4[p.sub.2][r.sub.2]; that is, for [absolute value of (x)], [absolute value of (y)] < 1,

[mathematical expression not reproducible]. (42)

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

[mathematical expression not reproducible], (43)

which is equivalent to

[R.sub.1](x) := [m.sub.1][x.sup.2] + [n.sub.1]x + [r.sub.1] [greater than or equal to] 0, (44)

where

[mathematical expression not reproducible]. (45)

If (36) holds, then [absolute value of ([n.sub.1])] [greater than or equal to] 2[absolute value of ([m.sub.1])] and [R.sub.1](x) [greater than or equal to] [m.sub.1] + [r.sub.1] - [absolute value of ([n.sub.1])], which is nonnegative from (35). On the other hand, if (38) holds, then [absolute value of ([n.sub.1])] < 2[absolute value of ([m.sub.1])], [R.sub.1](x) [greater than or equal to](4[m.sub.1][r.sub.1] - [n.sup.2.sub.1])/4[m.sub.1], and (37) implies [R.sub.1](x) [greater than or equal to] 0. Either case establishes (44).

Theorem 6. 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 c,k [member of] R with c [not equal to] 0 satisfying

[kappa] [greater than or equal to] (1 + B)(1+A)/4 (A - B)[absolute value of (c)]. (46)

Further let A, B, k, and c satisfy either

[mathematical expression not reproducible] (47)

whenever

[absolute value of (2[kappa] (1 - B) (A + B) c + [(1+ B).sup.2] (1 + A) c)] [greater than or equal to] - 1/2 (A - B)(1 - B)[c.sup.2] (48)

or the inequality

[mathematical expression not reproducible] (49)

when

[absolute value of (2[kappa] (1 - B) (A + B) c + [(1+ B).sup.2] (1 + A) c)] < 1/2(A - B)(1 - B)[c.sup.2]. (50)

If (1 + B)[U.sub.p](z)[not equal to] (1 + A) for all z [member of] D, then (-4k/c)[u'.sub.p] (z) [member of] P[A, B].

Theorem 7. Let 3 - 2[square root of 2 < B < A [less than or equal to] 1. Suppose c,[kappa] [member of] R and a [not equal to] 0, such that

[kappa] [greater than or equal to] (1 + B)(1 + A)/4(A - B) [absolute value of (c)]. (51)

Suppose A, B, [kappa], and c satisfy either

[mathematical expression not reproducible] (52)

whenever

[absolute value of (k [(1 + B).sup.3] (A + B) c + 8B (1 - [B.sup.2]) (1 + A) c)] [greater than or equal to] [c.sup.2]/4 (A - B) [(1 + B).sup.3] (53)

or the inequality

[mathematical expression not reproducible] (54)

when

[absolute value of (2[kappa][(1 + B).sup.3] (A + B) c + 8B (1 - [B.sup.2]) (1 + A) c)] < [c.sup.2]/4 (A - B)[(1 + B).sup.3].

If (1 + B)[U.sub.p](z) [not equal to] (1 + A) for all z [member of] D, then (-4[kappa]/c)[u'.sub.p](z) [member of] P[A,B].

Corollary 8. Let c [less than or equal to] -1 and

k [greater than or equal to] max {c(c + 1)/2}, c/(c + 1). (56)

Then (-4[kappa]/c)([u.sub.p](z) - 1) is close-to-convex of order (c + 1)/c with respect to the identity function.

Corollary 9. Let c be a nonzero real number and k [greater than or equal to] [absolute value of (c)]/2. Then

Re (-4[kappa]/c)[u'.sub.p] (z) > 1/2. (57)

3. Janowski Starlikeness of Generalized Bessel Functions

This section contributes to finding conditions to ensure a normalized and generalized Bessel function z[u.sub.p](z) in the class of Janowski starlike functions. For this purpose, first sufficient conditions for [u.sub.p]s (z) to be Janowski convex are determined, and then an application of relation (5) yields conditions for z[u.sub.p](z) [member of] [S.sup.*][A, B].

Theorem 10. Let c,[kappa] [member of] R be such that (A - B)[u'.sub.p](z) [not equal to] (1 + B) z[U".sub.p](z) for all z [member of] D and -1 [less than or equal to] B < A [less than or equal to] 1. Supposes

[kappa](1 + b) [greater than or equal to] [(1 + B).sup.2]/4(A - B))[absolute value of (c)] -(1 + A - B). (58)

Further let A, B, [kappa], and c satisfy

[mathematical expression not reproducible]. (59)

If 0 [not member of] [u'.sub.p](D) and 0 [not member of] [u".sub.p](D), then

1 + Z[u.sub."p](z)/[u'.sub.p](z) < 1 + Az/1 + Bz. (60)

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

p(z) := (A-B)[u'.sub.p] (z) + (1 - B)z[u".sub.p](z)/(A-B)[u'.sub.p](z)-(1 + B)z[u".sub.p](z). (61)

Then

[mathematical expression not reproducible] (62)

[mathematical expression not reproducible]. (63)

A rearrangement of (63) yields

z[u".sub.p](z)/[U".sub.p](z) = 2zp'(z)(p(z]-1)((p(z) + 1) + B(p(z)-1)) - 1 + z[u".sub.p](z)/u'p(z). (64)

Thus,

[mathematical expression not reproducible]. (65)

Now a differentiation of (4) leads to

4[z.sup.2][u'".sub.p](z) + 4 ([kappa] + 1) z[u".sub.p] (z) + cz[u'.sub.p] (z) = 0, (66)

which gives

[mathematical expression not reproducible] (67)

Using (62) and (65), (67) yields

[mathematical expression not reproducible], (68)

and equivalently

[mathematical expression not reproducible]. (69)

Define

[PHI][(p(z), zp' (z), z) := zp' (z) + [F.sub.1] (p (z)).sup.2] + [F.sub.2]p(z) + [F.sub.3],

where

[mathematical expression not reproducible], (71)

Thus, (69) yields [PHI](p(z),zp'(z),z) [member of] [OMEGA] = {0}. Now, with z = x + iy [member of] D,let

[mathematical expression not reproducible]. (72)

[mathematical expression not reproducible], (73)

Note that condition (58) 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 p 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. (74)

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

[mathematical expression not reproducible], (75)

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

H (x) = [h.sub.2] (A, B)x + [h.sub.3] (A, B) > 0, (76)

International Journal of Analysis

where

[mathematical expression not reproducible], (77)

Since [absolute value of (x)] < 1, the left-hand side of inequality (76) satisfies

[h.sub.2] (A,B)x + [h.sub.3] (A,B) [greater than or equal to] - [absolute value of ([h.sub.2])] (A,B) + [h.sub.3] (A,B). (78)

Now it is evident from (59) that H(x) [greater than or equal to] 0 which establishes inequality (76).

Thus [phi] satisfies the hypothesis of Lemma 1, and hence Re p(z) > 0, or equivalently

(A-B)[u'.sub.p] + (1-B)z[u".sub.p]/(A-B) [u'.sub.p] -(1 + B)z[u".sub.p] < 1+z/1-z (79)

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

(A-B)[u'.sub.p] (z) + (1- B) z[u'.sub.p] (z)/(A-B) [u'.sub.p] (z)-(1 + B) z[u".sub.p] 1+w(z)/(z) 1-w(z). (80)

A simple computation shows that

1 + z[u".sub.p] (z)/[u'.sub.p](z) = 1 + Aw(z)/1 + Bw(z), (81)

and hence

1 + Z[u".sub.p](z)/[u'.sub.p](z) < 1 + Az/1 + Bz. (82)

Relation (5) also shows that

z(z[u.sub.P] (z))'x/Z[U.sub.p](z) = 1 + [u".sub.p-1](z)/ [u'.sub.p-1](z). (83)

Together with Theorem 10, relation (83) immediately yields the following result for z[u.sub.p](z) [member of] [S.sup.*] [A, B].

Theorem 11. Let c and k be real numbers such that (A - B)[u'.sub.p-1] (z) = (1+B) z[U'.sub.p-1](z) for all z [member of] D and -1 < B < A [less than or equal to] 1.

Suppose

[kappa](1 + b)[greater than or equal to] [(1 + B).sup.2]/4(A - B) [absolute value of (c)] -(a-2B). (84)

Further let A, B, k, and c satisfy

[mathematical expression not reproducible]. (85)

Then z[u.sub.p](z) [member of] [S.sup.*][A,B].

http://dx.doi.org/10.1155/2016/4740819

Competing Interests

The authors declare that they have no competing interests.

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Saiful R. Mondal and Mohammed Al Dhuain

Department of Mathematics and Statistics, Collage of Science, King Faisal University, Hofuf, Al-Hasa 31982, Saudi Arabia

Correspondence should be addressed to Saiful R. Mondal; smondal@kfu.edu.sa

Received 19 April 2016; Accepted 4 October 2016