# Certain Geometric Properties of Generalized Dini Functions.

1. Introduction

Let A be the class of functions f of the form

f(z) = z + [[infinity].summation over (n=2)[a.sub.n][z.sup.n]], (1)

analytic in the open unit disc U = {z : [absolute value of z] < 1} and S denote the class of all functions in A which are univalent in U. Let [S.sup.*]([alpha]), C([alpha]), K([alpha]), and [??]([alpha]) denote the classes of star-like, convex, close-to-convex and strongly star-like functions of order a, respectively, and they are defined as

[mathematical expression not reproducible], (2)

[mathematical expression not reproducible]. (3)

It is clear that

[??] (1) = [S.sup.*] (0) = [S.sup.*],

C(0) = C

K(0) = K.

If f and g are analytic functions, then the function f is said to be subordinate to g, written as f(z) < g(z), if there exist a Schwarz function w which is analytic with w(0) = 0 and [absolute value of w] < 1 such that f(z) = g(w(z)). Furthermore, if the function g is univalent in U then we have the following equivalent relation:

f(z) < g(z) [??} f(0) = g(0), f(U) [subset] g (U). (5)

Special functions have great importance in pure and applied mathematics. The wide use of these functions has attracted many researchers to work on the different directions. Geometric properties of special functions such as hypergeometric functions, Bessel functions, Struve functions, Mittag-Leffler functions, Wright functions, and some other related functions are an ongoing part of research in geometric function theory. We refer to some geometric properties of these functions [1-10] and references therein.

Consider the second-order homogeneous differential equation

[z.sup.2]w" (z) + bzw (z) + (c[z.sup.2] - [v.sup.2] + (1 -b) v)w(z) = 0, (6)

where b, c, and v are complex numbers. The particular solution of the homogenous differential equation (6) is called the generalized Bessel functions of the first kind of order v. It is defined as

[mathematical expression not reproducible], (7)

where [GAMMA](*) denotes the gamma function. The function [w.sub.v,b,c] unifies the Bessel, modified Bessel, and spherical Bessel functions.

Special cases

(1) For b = c = 1, we have the Bessel functions of first kind of order v, defined as

[mathematical expression not reproducible]. (8)

(2) For b = 1, c = -1, we obtain the modified Bessel functions of first kind of order v whose series form is given as

[mathematical expression not reproducible]. (9)

(3) For b = 2, c = 1, we have the spherical Bessel functions of first kind of order v, given as

[mathematical expression not reproducible]. (10)

For more details about these functions, see [5, 11]. Recently, Deniz et al.  studied the function [[phi].sub.v,b,c] which is defined by the relation as

[mathematical expression not reproducible], (11)

where b,c, v [member of] C. Using the well-known Pochhammer symbol

[mathematical expression not reproducible], (12)

we obtain the series form of the function [[phi].sub.v,b,c] as

[mathematical expression not reproducible], (13)

where k = v + {b + 1}/2 [not equal to] 0,-1, -2, ....

Bessel functions are indispensable in many branches of mathematics and applied mathematics. Thus, it is important to study their properties in many aspects. We consider the generalized Dini functions [d.sub.v,a,b,c] : U [right arrow] C defined as

[mathematical expression not reproducible], (14)

where a [member of] [R.sup.+], b [member of] R, c [member of] C, and v [member of] R. For a = b = c =1, we obtain the normalized Dini function of order v of the form

[mathematical expression not reproducible]. (15)

By putting v = 1/2 and v = 3/2, we obtain the following particular cases of normalized Dini functions

[mathematical expression not reproducible]. (16)

Recently Baricz et al.  studied the close-to-convexity of Dini functions and some monotonicity properties and functional inequalities

for the modified Dini function are discussed in . Further some geometric properties of Dini functions are studied in .

This paper studies the Dini function [d.sub.v,a,b,c] given by the power series (14). We determine the conditions on parameters that ensure the Dini function to be star-like of order a, convex of order [alpha], and close-to-convex of order ((1 + [alpha])/2). We also study the convexity in the domain [U.sub.1/2] = {z: [absolute value of z] < 1/2}. Sufficient conditions on univalency of an integral operator defined by Dini function are also studied. We find the conditions on normalized Dini function to belong to the Hardy space [H.sup.p].

To prove our main results, we need the following lemmas.

Lemma 1 (see ). If f [member of] A satisfy [absolute value of f'(z) - 1] < 1 for each z [member of] U, then f is convex in [U.sub.1/2] = {z : [absolute value of z] < 1/2}.

Lemma 2 (see ). Let [beta] [member of] C with Re([beta]) > 0, and c [member of] C with [absolute value of c] <1, c [not equal to] -1. If h [member of] A satisfies

[mathematical expression not reproducible], (17)

then the integral operator

[mathematical expression not reproducible], (18)

is analytic and univalent in U.

Lemma 3 (see ). Let G(z) be convex and univalent in the open unit disc with condition G(0) = 1. Let F(z) be analytic in the open unit disc with condition F(0) = 1 and F < G in the open unit disc. Then, Vn [member of] N U {0}, we obtain

[mathematical expression not reproducible]. (19)

Lemma 4 (see ). If f [member of] A satisfies the inequality

[absolute value of zf" (z)] < 1 - [alpha]/4, (z [member of] U, 0 [less than or equal to] [alpha] < 1), (20)

then

Ref' (z) < 1 - [alpha]/4, (z [member of] U, 0 [less than or equal to] [alpha] < 1), (21)

Lemma 5 (see ). If the function f(z) = z+[a.sub.2][z.sup.2] + ... + [a.sub.n][z.sup.n] + ... is analytic in U and in addition 1 [greater than or equal to] 2[a.sub.2] [greater than or equal to] ... [greater than or equal to] n[a.sub.n] ... [greater than or equal to] 0 or 1 [less than or equal to] 2[a.sub.2] [less than or equal to] ... [less than or equal to] n[a.sub.n] ... [less than or equal to] 2, then f is close-to-convex function with respect to the convex function z [right arrow] -log(1 - z). Moreover, if the odd function g(z) = z+[b.sub.3][z.sup.3] + ... + [b.sub.2n-1][z.sup.2n-1] + ... is analytic in U and if 1 [greater than or equal to] 3[b.sub.3][greater than or equal to] ... [greater than or equal to] (2n + 1)[b.sub.2n+1] ... [greater than or equal to] 0 or 1 [less than or equal to]3[b.sub.3] [less than or equal to] ... [less than or equal to] (2n + 1)[b.sub.2n+1] ... [less than or equal to] 2, then g is univalent in U.

2. Geometric Properties of Normalized Dini Functions

Theorem 6. Let a [member of] [R.sup.+], b [member of] R, c [member of] C, and v [member of] R with a [member of] [0,1) and z [member of] U. Then the following assertions are true:

(i) [mathematical expression not reproducible].

(ii) [mathematical expression not reproducible].

(iii) [mathematical expression not reproducible].

(iv) [mathematical expression not reproducible].

Proof. We use the inequality [absolute value of z[d'.sub.v,a,b,c](z)/[d.sub.v,a,b,c](z)-1] < 1-[alpha], to prove the starlikeness of order [alpha] for the function [d.sub.v,a,b,c]. So by using the well-known triangle inequality

[absolute value of [z.sub.1] + [z.sub.2]] [less than or equal to] [absolute value of [z.sub.1]] + [absolute value of [z.sub.2]], (22)

with the equality

[mathematical expression not reproducible], (23)

and the inequalities

[mathematical expression not reproducible], (24)

we obtain

[mathematical expression not reproducible]. (25)

Furthermore, if we use reverse triangle inequality

[absolute value of [z.sub.1] + [z.sub.2]] [less than or equal to][parallel][z.sub.1][absolute value of -][z.sub.2][parallel], (26)

and the inequalities

[mathematical expression not reproducible], (27)

then we get

[mathematical expression not reproducible] (28)

By combining inequalities (25) and (28), we get

[mathematical expression not reproducible]. (29)

So, [d.sub.v,a,b,c] is star-like function of order [alpha], where [mathematical expression not reproducible].

(ii) To prove the convexity of order a of function [d.sub.v,a,b,c], we have to show that [absolute value of z[d".sub.v,a,b,c](z)[d'.sub.v,a,b,c](z)] < 1 - [alpha]. By using the well-known triangle inequality

[absolute value of [z.sub.1] + [z.sub.2]][less than or equal to][absolute value of [z.sub.1]] + [absolute value of [z.sub.2], (30)

with equality

[mathematical expression not reproducible]. (31)

and the inequalities

[mathematical expression not reproducible]. (32)

we have

[mathematical expression not reproducible]. (33)

Moreover, if we use reverse triangle inequality

[absolute value of [z.sub.1] + [z.sub.2]][less than or equal to][parallel][absolute value of -]][z.sub.2][parallel], (30)

with equality

[mathematical expression not reproducible]. (35)

and the inequalities

[mathematical expression not reproducible], (36)

we get

[mathematical expression not reproducible]. (37)

By combining inequalities (33) and (37), we have

[mathematical expression not reproducible]. (38)

This implies that [d.sub.v,a,b,c] is convex function of order a, where [mathematical expression not reproducible].

(iii) Using inequality (33) and Lemma 4, we have

[mathematical expression not reproducible], (39)

where [mathematical expression not reproducible]. This shows that [d.sub.v,a,b,c][member of] K((1 + [alpha])/2). Therefore Re([d'.sub.v,a,b,c](z)) > (1 + [alpha])/2.

(iv) To prove that [d.sub.v,a,b,c]/z [member of] P([alpha]), we have to show that [absolute value of g(z) - 1] < 1, where g(z) = ([d.sub.v,a,b,c](z)/(z - [alpha]))/(1 - [alpha]). By using the well-known triangle inequality

[absolute value of [z.sub.1] + [z.sub.2]][less than or equal to][absolute value of [z.sub.1]]] + [absolute value of z.sub.2]], (40)

with the equality

[mathematical expression not reproducible], (41)

and the inequality

(n - 1)![(k+1).sub.n-1] [greater than or equal to][(k+1).sup.n-1], n[member of] N, (42)

we have

[mathematical expression not reproducible]. (44)

Therefore, [d.sub.v,a,b,c]/z [member of] P([alpha]) for 0 < [alpha] < 1 - (2[absolute value of c](k + 1) + A [absolute value of c](k+ 1))/(ak(4(k + 1) - [absolute value of c])).

Corollary 7. By using Theorem 6 assertions (i) and (iv), we get the following corollary:

(i) If 0 [less than or equal to][alpha] < [[alpha].sub.1] where [[alpha].sub.1] [equivalent] 0.333333333..., then [d.sub.1/2][member of] [S.sup.*]([alpha]).

(ii) If 0 [less than or equal to] [alpha] < [[alpha].sub.2] where [[alpha].sub.2] [equivalent] 0.7727272727 ..., then [d.sub.3/2][member of] [S.sup.*]([alpha]).

(ii) If 0 [less than or equal to] [alpha] < [[alpha].sub.3] where [[alpha].sub.3] [equivalent] 0.44444444 ..., then [d.sub.1/2]/z[member of]P([alpha]).

(iii) [less than or equal to] [alpha] < [[alpha].sub.4] where [[alpha].sub.4] [equivalent] 0.44444444 ..., then [d.sub.3/2]/z[member of]P([alpha]).

Putting [alpha] = 0 in Theorem 6, we have the following results.

Corollary 8. Let a [member of] [R.sup.+], b [member of] R, c [member of] C, and v [member of] R. Then the following assertions are true.

(i) [mathematical expression not reproducible].

(ii) [mathematical expression not reproducible].

(iii) [mathematical expression not reproducible].

(iv) [mathematical expression not reproducible].

Corollary 9. By putting a = b = c = 1 and k = v +1 in

Corollary 8, we obtain the following results:

(i) If v > 0.453768227, then [d.sub.v] [member of] [S.sup.*].

(ii) If v > 2.406035949, then [d.sub.v] [member of] C.

(iii) If v > 5.548088707, then [d.sub.v] [member of] K(1/2).

(iv) If v > -0.8660254038, then [d.sub.v]/z [member of] P.

Remark 10. It is easy to see that our results improve the results of  as special cases of parameters. Some particular cases regarding Corollary 9 are given below.

Corollary 11. (i) If v = 1/2, then [d.sub.1/2](z) = z cos [square root of z] [member of] [S.sup.*].

(ii) If v = 3/2, then [d.sub.3/2](z) = 3 cos [square root of z] - (3(z - 2) sin [square root of z])/2/z [member of] [S.sup.*].

(iii) If v = 1/2, then ([d.sub.1/2]/z)(z) = cos [square root of z] [member of] P.

(iv) If v = 3/2, then [d.sub.3]2](z)/z = (6[square root of z] cos [square root of z] - 3(z -2)sin/z)/2[z.sup.3/2] [member of] P.

Theorem 12. Let a [member of] [R.sup.+], b [member of] R, c [member of] C, and v [member of] R. If ((4(k + 1) - [absolute value of c])([absolute value of c](a + 4)) + [[absolute value of c].sup.2](a + 6) + 4a[absolute value of c](k + 1))/(4ak(4(k + 1) - [absolute value of c])) < 1, then [d.sub.v,a,b,c] is convex in [U.sub.1/2].

Proof. By using the well-known triangle inequality

[absolute value of [z.sub.1] + [z.sub.2]] [less than or equal to][absolute value of [z.sub.1]] + [z.sub.2], (44)

with the equality [GAMMA](v + 1)/[GAMMA](v+n+1) = 1/((v + 1)(v+2) ...(v+n)) = 1/[(v + 1).sub.n], n [member of] N, and the inequalities

[mathematical expression not reproducible], (45)

we obtain

[mathematical expression not reproducible], (46)

In view of Lemma 1, [d.sub.v,a,b,c] is convex in [U.sub.1/2], if ((4(k + 1) - [absolute value of c])([absolute value of c](fl+4))+[absolute value of c]2(fl+6)+4fl[absolute value of c](k+1))/(4flk(4(k+1)-[absolute value of c]))< 1, but this is true under the hypothesis.

Remark 13. When we put a = b = c = 1, our results are better than .

Consider the integral operator [F.sub.[beta]] : U [right arrow] C, where [beta][member of] C, [beta] [not equal to] 0,

[mathematical expression not reproducible]. (47)

Here [F.sub.[beta]] [member of] A. In the next theorem, we obtain the conditions so that [F.sub.[beta]] is univalent in U.

Theorem 14. Let a [member of] [R.sup.+], b [member of] R, c [member of] C, and v [member of] R. Let (2[absolute value of c] + a[absolute value of c](4(k + 1) - [absolute value of c]) + 4[[absolute value of c].sup.2] + a[[absolute value of c].sup.2])/(-[[absolute value of c].sup.2](2 + a) + (4(k + 1)-[absolute value of c])(4ak-2[absolute value of c]-a[absolute value of c])) < 1 and suppose that M is a positive real number such that [absolute value of [F.sub.[beta]](z)] < M in the open unit disc. If

[mathematical expression not reproducible], (48)

then [F.sub.[beta]] is univalent in U.

Proof. A calculation gives us

[mathematical expression not reproducible], (48)

Since [d.sub.v,a,b,c][member of] A, then, by the Schwarz Lemma, triangle inequality, and (29), we obtain

[mathematical expression not reproducible]. (50)

This shows that the given integral operator satisfies Becker's criterion for univalence, and hence [F.sub.[beta]] is univalent in U.

3. Strong Starlikeness

In this section, we are mainly interested about some sufficient conditions under which the normalized Dini function belongs to the class of strong starlikeness of order a.

Theorem 15. Let a [member of] [R.sup.+], b [member of] R, c [member of] C, and v [member of] R. If ((4(k + 1) - [absolute value of c])([absolute value of c](a + 4)) + [[absolute value of c].sup.2](a + 6) + 4a[absolute value of c](k+ 1))/(4ak(4(k + 1) - [absolute value of c])) < 1, then [d.sub.v,a,b,c][member of] [[??].sub.*]([alpha]), where

[mathematical expression not reproducible], (51)

and x = ((4(k + 1) - [absolute value of c])([absolute value of c](a + 4)) + [[absolute value of c].sup.2](a + 6) + 4a[absolute value of c](k + 1))/(4ak(4(k+1) - [absolute value of c])).

Proof. By using the well-known triangle inequality

[absolute value of [z.sub.1 + [z.sub.2]][less than or equal to][absolute value of [z.sub.1]] + [absolute value of [z.sub.2]], (52)

with the equality [GAMMA](v+1)/[GAMMA](v+n+1) = 1/((v+1)(v+2) ... (v + n)) = 1/[(v + 1).sub.n], n [member of] N, and the inequalities

[mathematical expression not reproducible], (53)

we obtain

[mathematical expression not reproducible]. (54)

For 0< x [less than or equal to] 1 and from (54), we concluded that

[mathematical expression not reproducible]. (56)

With the help of Lemma 3, take n =0 with F(z) = [d'.sub.v,a,b,c] (z) and G(z) = 1 + xz, and we get

[mathematical expression not reproducible]. (57)

As a result

[mathematical expression not reproducible]. (58)

By using (56) and (58), we obtain

[mathematical expression not reproducible], (59)

which implies that [mathematical expression not reproducible].

Corollary 16. For k = [k.sup.*], where [k.sup.*] is the positive root of 2[k.sup.2] - 3 [square root of 5k] - 2[square root of 5] = 0, then [d.sub.v,a,b,c](z) [member of] [S.sup.*].

4. Close-to-Convexity with respect to Certain Functions

Recently many authors discuss the close-to-convexity of some special functions with respect to certain functions. Here, we are also interested in the close-to-convexity of normalized Dini function.

Theorem 17. Let a [member of] [R.sup.+], b [member of] R, c [member of] R, and v [member of] R. If [GAMMA](v + n + 1) [greater than or equal to] - c(2n + a)[GAMMA](v + n)/(2(n - 1) + a), then [d.sub.v,a,b,c] is close-to-convex with respect to -log(1 - z).

Proof. Set

[mathematical expression not reproducible]. (60)

To prove that f is close-to-convex with respect to the function -log(1-z), we use Lemma 5. Therefore, we have to prove that [{[nb.sub.n-1]}.sub.n[greater than or equal to]2] is a decreasing sequence. After some computations, we obtain

[mathematical expression not reproducible]. (61)

By using the conditions on parameters, we easily observe that [nb.sub.n-1] - (n +l))[b.sub.n] > 0 for all n [greater than or equal to] 2, and thus [{[nb.sub.n-1]}.sub.n[greater than or equal to]2] is a decreasing sequence. By Lemma 5 it follows that f is close-to-convex with respect to the function - log(1 - z).

Theorem 18. Let a [member of] [R.sup.+], be R, c [member of] R, and v [member of] R. If [GAMMA](v + n +1) [greater than or equal to] -c(2n + a)[GAMMA](v + n)/(2(n - 1) + a), then [d.sub.v,a,b,c] is close-to-convex with respect to (1/2)log((1 + z)/(1 - z)).

Proof. Set

[mathematical expression not reproducible]. (62)

Here [B.sub.2n-1] = [b.sub.n-1] = [(-c).sup.n-1] (2(n - 1) + a)[GAMMA](v + 1)/a[4.sup.n-1](n - 1)![GAMMA] (v + n +1), and therefore we have [b.sub.1] = -c(2 + a)/4ak and [B.sub.2n-1] > 0 for all n [greater than or equal to] 2. To prove our main result we will prove that [{(2n - 1)[b.sub.n-1]}.sub.n[greater than or equal to]2] is a decreasing sequence. After some computations, we obtain

[mathematical expression not reproducible]. (63)

By using the conditions on parameters we easily observe that (2n - 1)[b.sub.n-1] - (2n +l)[b.sub.n] > 0 for all n [greater than or equal to] 2, and thus [{(2n - 1)[b.sub.n-1]}.sub.n[greater than or equal to]2] is a decreasing sequence. By Lemma 5 it follows that f(z) is close-to-convex with respect to the function (1/2) log((1 + z)/(1 - z)).

5. Hardy Spaces of Dini Function

Let H denote the class of all analytic functions in the open unit disk U ={z : [absolute value of z] <1} and [H.sup.[infinity]] denote the space of all bounded functions on H. This is a Banach algebra with respect to the norm

[mathematical expression not reproducible]. (64)

We denote [H.sup.p], 0 < p < [infinity] for the space of all functions f [member of] H such that [[absolute value of f].sup.p] admits a harmonic majorant. [H.sup.p] is a Banach space if the norm of f is defined to be p-th root of the least harmonic majorant of [[absolute value of f].sup.p] for some fixed z [member of] U. Another equivalent definition of norm is given as follows: Let f [member of] H, and set

[mathematical expression not reproducible]. (65)

Then the function f [member of] [H.sup.p] if [M.sub.p](r, f) is bounded for all r [member of] [0,1). It is clear that

[H.sup.[infinity]] [.sub.c] [H.sup.q] [subset] [H.sup.p], 0 < q < p < [infinity]. (66)

For some details see . It is also known  that, for Re(f'(z)) > 0 in U, then

f' [member of] [H.sup.q], q < 1, f [member of][H.sup.q/(1-q)], 0 < q < 1. (67)

Hardy spaces of hypergeometric functions are recently studied by Ponnusamy . Baricz  uses the idea of Ponnusamy and found the Hardy spaces of Bessel functions while Yagmur and Orhan  studied the same problem for generalized Struve functions. Similarly Yagmur  studied the problem for Lommel functions and Raza et al.  studied the same problem for Wright functions.

To prove our main results we need the following lemmas.

Lemma 19 (see ). [P.sub.0]([alpha]) * [P.sub.0]([beta])[subset] [P.sub.0]([gamma]), where [gamma] = 1 - 2(1 - [alpha])(1 - [beta]) with [alpha], [beta] < 1 and the value of [gamma] is best possible.

Lemma 20 (see ). For [alpha], [beta] < 1 and [gamma] = 1 - 2 (1 - [alpha])(1-[beta]), we have [R.sub.0]([alpha]) * [R.sub.0]([beta]) [subset] [R.sub.0]([gamma]) or equivalently [P.sub.0]([alpha]) * [P.sub.0]([beta]) [subset] [P.sub.0]([gamma]).

Lemma 21 (see ). If the function f, convex of order [alpha], where [alpha] [member of] [0,1), is not of the form

[mathematical expression not reproducible], (68)

for some complex numbers [zeta] and [eta] and for some real number [gamma] then the following statements hold:

(i) There exists [delta] = [delta](f) > 0, such that f' [member of] [H.sup.[delta]+1/2](1 - [alpha]).

(ii) If [alpha] [member of] [1,1/2), then there exists [tau] = [tau](f) > 0, such that f [member of] [H.sup.[tau]+1]/(1-2[alpha])

(iii) If [alpha][greater than or equal to] 1/2, then f [member of] [H.sup.[infinity]].

The lemma defined below also plays an important role to find our main results.

Theorem 22. Let [mathematical expression not reproducible]. Then

(i) [d.sub.v,a,b,c][member of] [H.sup.1/1-2[alpha]] for [alpha][member of][0,1/2);

(ii) [d.sub.v,a,b,c][member of] [H.sup.[infinity]] for [alpha][greater than or equal to] 1/2.

Proof. By using the definition of Hypergeometric function

[mathematical expression not reproducible], (69)

we have

[mathematical expression not reproducible], (70)

for [zeta], [eta][member of] C, and [alpha] [not equal to] 1/2 and for [gamma][member of] R. On the other hand

[mathematical expression not reproducible]. (71)

This implies that [d.sub.v,a,b,c] is not of the form [zeta] + [eta]z[(1 - z[e.sup.i[gamma]]).sup.2a[alpha]-1] for [alpha] = 1/2 and [zeta] + [eta]log [(1 - z[e.sup.i[gamma]] for [alpha] = 1/2, respectively. Also, from part (ii) of Theorem 6, [d.sub.v,a,b,c] is convex of order a. Hence, by using Lemma 21, we have required result.

Theorem 23. Let v > -0.8660254038 and f [member of] R. Then the convolution [d.sub.v] * f is in [H.sup.[infinity]] [intersection] R.

Proof. Let h(z) = [d.sub.v] * f(z). Then it is clear that h'(z) = [d.sub.v]/[z.sup.*] f'(z). Using Corollary 9 of part (iv), we have [d.sub.v]/z [member of] P. Since f [member of] R, therefore by using Lemma 19 h [member of] R. It is also clear that [d.sub.v]/z is an entire function and therefor h is entire. This implies that h is bounded. Hence, we have the required result.

Theorem 24. Letv [member of] R with (2[absolute value of c](k+1)+a[absolute value of c](k+1))/ ak(4(k+1) - [absolute value of c]) < 1 - [alpha], and then [d.sub.v,a,b,c]/z [member of] P([alpha]), [alpha] [member of] [0,1), and z [member of] U. If f [member of] R([beta]), with [beta] < 1, then [d.sub.v,a,b,c] * f [member of] R([gamma]), where [gamma] = 1 - 2(1 - [alpha])(1 - [beta]).

Proof. Let h(z) = [d.sub.v,a,b,c](z) * f(z), and then h'(z) = [d.sub.v,a,b,c](z)/z * f'(z). Now, from Theorem 6 of part (iv), we have [d.sub.v,a,b,c]/z [member of] P([alpha]). By using Lemma 19 and the fact that f' [member of] P([beta]), we have h'(z) [member of] P([gamma]), where [gamma] = 1-2(1-[alpha])(1-[beta]). Consequently, we have h [member of] R([gamma]).

Corollary 25. Let 0 [less than or equal to][alpha] < 1, (2[absolute value of c](k+1)+a[absolute value of c](k+1))/ak(4(k+1) - [absolute value of c]) < 1 - [alpha], and then [d.sub.v,a,b,c]/z [member of] P([alpha]). If f [member of] R([beta]), [beta] = (1 - 2[alpha])(2 - 2[alpha]), then [d.sub.v,a,b,c] * f [member of] R(0).

Corollary 26. Let v > -0.8660254038. If f [member of] R(1/2), then [d.sub.v] * f [member of] R(0).

By using Theorem 23 and Corollary 25, then we have the following corollary.

Corollary 27. If f [member of] R, then the convolutions [d.sub.1/2] * f and [d.sub.3/2] * f are in [H.sup.[infinity]][intersection] R. Moreover, if f [member of] R(1/2), then [d.sub.1/2] * f and [d.sub.3/2] * f [member of] R(0).

https://doi.org/10.1155/2018/2684023

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this paper.

Acknowledgments

The work here is supported by UKM Grant no. GUP-2017064.

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Muhey U. Din, (1) Mohsan Raza (iD),(1) Saqib Hussain (iD),(2) and Maslina Darus (3)

(1) Department of Mathematics, Government College University, Faisalabad, Pakistan

(3) Faculty of Science and Technology, School of Mathematical Sciences, Universiti Kebangsaan Malaysia, 43600 Bangi, Selangor, Malaysia

Correspondence should be addressed to Mohsan Raza; mohsan976@yahoo.com

Received 20 March 2018; Accepted 25 June 2018; Published 11 July 2018