New Subclasses concerning Some Analytic and Univalent Functions.

1. Introduction and Preliminaries

Let A be the class of functions f(z) which are analytic in the open unit disk U = {z [member of] C : [absolute value of z] < 1} with f(0) = 0 and f'(0) = 1.

Let S denote the subclass of A consisting of functions f(z) [member of] A which are univalent in U. Also, let [S.sup.*]([beta]) be the subclass of S consisting of f(z) which are starlike of order [beta] (0 [less than or equal to] [beta] < 1) in U. Further, we say that f(z) [member of] K([beta]) if f(z) [member of] S satisfies zf '(z) [member of] [S.sup.*] ([beta]). A function f(z) [member of] K([beta]) is said to be convex of order [beta] in U (cf. [1-3]).

With the above definitions for classes K([beta]), [S.sup.*] ([beta]), S, and A, it is known that

K ([beta]) [subset] [S.sup.*] ([beta]) [subset] S [subset] A (1)

and f(z) [member of] [S.sup.*]([beta]) if and only if [[integral].sup.z.sub.0] (f(t)/t)dt [member of] K([beta]). The function f(z) given by

f(z) = z/1 - [z.sup.2] = z + [z.sup.3] + [z.sup.5] + ... (z [member of] U) (2)

is in the class [S.sup.*](0) [equivalent to] [S.sup.*] and the function f(z) given by

f(z) = z/1 - z = z + [z.sup.2] + [z.sup.3] + ... (z [member of] U) (3)

is in the class K(0) [equivalent to] K.

If we consider the function f(z) given by

[f.sub.[alpha]](z) = z/1 - [z.sup.[alpha]] = z + [[infinity].summation over (n=1)] [z.sup.1+n[alpha]] (z [member of] U) (4)

for some real [alpha] (0 < [alpha] [less than or equal to] 2), we discuss some properties between functions f(z) in (2) and (3), where we consider the principal value for [z.sup.n[alpha]].

With the function f(z) given by (4), we introduce a class [A.sub.[alpha]] of analytic functions f(z) with series expansion in U such that

f(z) = z + [[infinity].summation over (n=1)] [a.sub.n][z.sup.1+n[alpha]] (z [member of] U) (5)

for some real [alpha] (0 < [alpha] [less than or equal to] 2), where we take the principal value for [z.sup.n[alpha]]. If f(z) [member of] [A.sub.[alpha]] satisfies

Re(zf'(z)/f(z)) > [beta] (z [member of] U) (6)

for some real [beta] (0 [less than or equal to] [beta] < 1), then we say that f(z) [member of] [S.sup.*.sub.[alpha]] ([beta]).

Also, if f(z) [member of] [A.sub.[alpha]] satisfies

Re(1 + zf"(z)/f'(z)) > [beta] (z [member of] U) (7)

for some real [beta] (0 [less than or equal to] [beta] < 1), then we say that f(z) [member of] [K.sub.[alpha]]([beta]).

With the above definitions for the classes [S.sup.*.sub.[alpha]]([beta]) and [K.sub.[alpha]]([beta]), we have that f(z) [member of] [K.sub.[alpha]]([beta]) if and only if zf'(z) [member of] [S.sup.*.sub.[alpha]]([beta]) and that f(z) [member of] [S.sup.*.sub.[alpha]]([beta]) if and only if [[integral].sup.z.sub.0](f(t)/t)dt [member of] [K.sub.[alpha]]([beta]).

2. Some Properties

In this section, we consider some properties of functions with series expansion given by (4).

Theorem 1. If f(z) is given by (4), then f(z) [member of] [S.sup.*.sub.[alpha]]((2 - [alpha])/2) for 0 < [alpha] [less than or equal to] 2 and f(z) [member of] [K.sub.[alpha]]([alpha]) for 0 < [alpha] < 1.

Proof. For f(z) given by (4), we see that zf'(z)/f(z) = 1 for z = 0 and

[mathematical expression not reproducible] (8)

for z = [e.sup.i[theta]] (0 < [theta] < 2[pi]). This shows that f(z) [member of] [S.sup.*.sub.[alpha]]((2-[alpha])/2) for 0 < [alpha] [less than or equal to] 2. Further, we have that 1 + zf" (z)/f'(z) = 1 for z = 0 and

[mathematical expression not reproducible] (9)

for z = [e.sup.i[theta]] (0 < [theta] < 2[pi]). Letting

g(t) = 1 + ([alpha] - 1)t/1 + [([alpha] - 1).sup.2] + 2 ([alpha] - 1)t (t = cos([alpha][theta])), (10)

we have that

[mathematical expression not reproducible]. (11)

Thus, we see that

Re (1 + zf"(z)/f'(z)) > [alpha] (z [member of] U) (12)

for 0 < [alpha] < 1. This completes the proof of the theorem.

Corollary 2. A function

f(z) = z/1 - [square root of (z)] (z [member of] U) (13)

belongs to the class [S.sup.*.sub.1/2](3/4) and [K.sub.1/2](1/2).

Next, we discuss some properties of functions f(z) for [A.sub.[alpha]].

Theorem 3. If f(z) given by (5) satisfies

[[infinity].summation over (n=1)] (n[alpha] + 1 - [beta])[absolute value of [a.sub.n]] [less than or equal to] 1 - [beta]] (14)

for some [beta] (0 [less than or equal to] [beta] < 1), then f(z) [member of] [S.sup.*.sub.[alpha]]([beta]).

The equality holds true for f(z) given by

[mathematical expression not reproducible]. (15)

Proof. Let the function f(z) be given by (5); then, we have that

[mathematical expression not reproducible] (16)

if f(z) satisfies (14). This shows that f(z) [member of] [S.sup.*.sub.[alpha]]([beta]). Further, if we consider a function f(z) given by (15), then we see that

[mathematical expression not reproducible]. (17)

Theorem 4. If f(z) given by (5) satisfies

[[infinity].summation over (n=1)](n[alpha] + 1)(n[alpha] + 1 - [beta]) [absolute value of [a.sub.n]] [less than or equal to] 1 - [beta] (18)

for some [beta] (0 [less than or equal to] [beta] < 1), then f(z) [member of] [K.sub.[alpha]]([beta]).

The equality in (18) holds true for f(z) given by

[mathematical expression not reproducible]. (19)

Further, we obtain the following.

Theorem 5. Let f(z) be given by (5) with arg [a.sub.n] = [pi] - n[alpha][theta] (0 < [theta] < 2[pi]). Then, f(z) [member of] [S.sup.*.sub.[alpha]]([beta]) ifand onlyif

[[infinity].summation over (n=1)] (n[alpha] + 1 - [beta]) [absolute value of [a.sub.n]] [less than or equal to] 1 - [beta] (20)

for some [beta] (0 [less than or equal to] [beta] < 1). The equality holds true for

[mathematical expression not reproducible]. (21)

Proof. Theorem 3 implies that if f(z) satisfies (20), then f(z) [member of] [S.sup.*.sub.[alpha]]([beta]). Next, we suppose that f(z) [member of] [S.sup.*.sub.[alpha]]([beta]). Then,

[mathematical expression not reproducible]. (22)

If we consider z = r[e.sup.i[theta]], then we have that

[a.sub.n][z.sup.n[alpha]] = [absolute value of [a.sub.n]][r.sup.n[alpha]][e.sup.i[pi]] = -[absolute value of [a.sub.n]][r.sup.n[alpha]]. (23)

Then, we obtain that

[mathematical expression not reproducible]. (24)

This gives us

[mathematical expression not reproducible], (25)

that is,

[[infinity].summation over (n=1)] (n[alpha] + 1 - [beta])[absolute value of [a.sub.n]] [less than or equal to] 1 - [beta]. (26)

Thus, f(z) [member of] [S.sup.*.sub.[alpha]]([beta]) if and only if the coefficient inequality (20) holds true.

Further, for the class [K.sub.[alpha]]([beta]), we have the following.

Theorem 6. Let f(z) be given by (5) with arg [a.sub.n] = [pi] - n[alpha][theta] (0 < [theta] < 2[pi]). Then, f(z) [member of] [K.sub.[alpha]]([beta]) if and only if

[[infinity].summation over (n=1)] (n[alpha] + 1)(n[alpha] + 1 - [beta]) [absolute value of [a.sub.n]] [less than or equal to] 1 - [beta] (27)

for some [beta] (0 [less than or equal to] [beta] < 1). The equality holds true for

[mathematical expression not reproducible]. (28)

In this section, we consider

d(z) = z/1 - [z.sup.[alpha]] (z [member of] U) (29)

for some real [alpha] > 2. Then, we say that g(z) [not member of] [S.sup.*.sub.[alpha]]([beta]) and g(z) [not member of] [K.sub.[alpha]]([beta]) for any real [beta] (0 [less than or equal to] [beta] < 1). Now, we derive the following.

Theorem 7. If g(z) is given by (29) with [alpha] > 2, then

Re(zg'(z)/g(z)) > 1 - ([alpha] - 1)[r.sup.[alpha]]/1 + [r.sup.[alpha]] (0 < [absolute value of (z)] = r < 1). (30)

Proof. For g(z) given by (29), we have that

[mathematical expression not reproducible] (31)

for z = r[e.sup.i[theta]] [member of] U. This gives us

[mathematical expression not reproducible]. (32)

Letting

h(t) = 1 + ([alpha] - 2)[r.sup.[alpha]]t - ([alpha] - 1)[r.sup.2[alpha]]/1 + [r.sup.2[alpha]] - 2[r.sup.[alpha]]t (t = cos [alpha][theta]), (33)

we see that h'(t) > 0. This gives us

Re(zg'(z)/g(z)) > 1 - ([alpha] - 1)[r.sup.[alpha]]/1 + [r.sup.[alpha]]. (34)

Corollary 8. If g(z) is given by (29) with [alpha] > 2, then

[mathematical expression not reproducible] (35)

Proof. If we consider

Re(zg'(z)/g(z)) > 1 - ([alpha] - 1)[r.sup.[alpha]]/1 + [r.sup.[alpha]] [greater than or equal to] [beta], (36)

then

0 < r [less than or equal to] [alpha][square root of (1 - [beta]/[beta] + [alpha] - 1)] < 1. (37)

Remark 9. If [beta] = 0 in (35), then

0 < [absolute value of z] [less than or equal to] [alpha][square root of (1/[alpha] - 1)] < 1, (38)

and if [beta] = 1/2, then

0 < [absolute value of z] [less than or equal to] [alpha][square root of (1/2[alpha] - 1)] < 1. (39)

4. Partial Sums

Finally, we consider the partial sums of f(z) given by (5). In view of (5), we write

[f.sub.n](z) = z + [a.sub.n][z.sup.1+n[alpha]] (n = 1,2,3, ...) (40)

for some real [alpha] (0 < [alpha] [less than or equal to] 2). Recently, Darus and Ibrahim [4] and Hayami et al. [5] have shown some interesting results for some partial sums of analytic functions.

Now, we derive the following.

Theorem 10. Let [f.sub.n](z) be given by (40) with [absolute value of [a.sub.n]] [less than or equal to] 1. Then,

Re(z[f'.sub.n](z)/[f.sub.n](z)) > 1 - (n[alpha] + 1)[absolute value of [a.sub.n]]/1 - [absolute value of [a.sub.n]] (z [member of] U), (41)

Re(z[f'.sub.n](z)/[f.sub.n](z)) [greater than or equal to] 1 - (n[alpha] + 1)[r.sup.n[alpha]]/1 - [r.sup.n[alpha]] ([absolute value of z] = r < 1). (42)

Proof. It follows that

[mathematical expression not reproducible]. (43)

where [a.sub.n] = [absolute value of [a.sub.n]][e.sup.i[phi]] and z = r[e.sup.i[theta]]. This gives us

[mathematical expression not reproducible]. (44)

Defining h(t) by

[mathematical expression not reproducible], (45)

we have that h'(t) > 0 with [absolute value of [a.sub.n]] [less than or equal to] 1. Thus, we obtain

[mathematical expression not reproducible]. (46)

Making r [right arrow] 1 in (46), we see (41). Also letting [absolute value of [a.sub.n]] = 1 in (46), we see (42).

Corollary 11. Let [f.sub.n](z) be given by (40) with [absolute value of [a.sub.n]] [less than or equal to] (1 - [beta])/(n[alpha] + 1 - [beta])(0 [less than or equal to] [beta] < 1). Then, [f.sub.n](z) [member of] [S.sup.*.sub.[alpha]]([beta]).

Proof. Since [absolute value of [a.sub.n]] < 1, [f.sub.n](z) satisfies (41).

Therefore, for [absolute value of [a.sub.n]] [less than or equal to] (1 - [beta])/(n[alpha] + 1 - [beta]), (41) gives us [f.sub.n](z) [member of] [S.sup.*.sub.[alpha]]([beta]).

https://doi.org/10.1155/2017/4674782

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 MOHE Grant FRGS/1/2016/ STG06/UKM/01/1.

References

[1] P. L. Duren, Univalent Functions, vol. 259, Springer-Verlag, New York, Berlin, Heidelberg, Tokyo, 1983.

[2] A. W. Goodman, Geometric Theory of Functions, vol. I and II, Mariner, Tampa, Fla, USA, 1983.

[3] M. I. S. Robertson, "On the theory of univalent functions," Annals of Mathematics. Second Series, vol. 37, no. 2, pp. 374-408, 1936.

[4] M. Darus and R. W. Ibrahim, "Partial sums of analytic functions of bounded turning with applications," Computational & Applied Mathematics, vol. 29, no. 1, pp. 81-88, 2010.

[5] T. Hayami, K. Kuroki, E. Y. Duman, and S. Owa, "Partial sums of certain univalent functions," Applied Mathematical Sciences, vol. 6, no. 13-16, pp. 779-805, 2012.

Maslina Darus (1) and Shigeyoshi Owa (2)

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

(2) Department of Mathematics, Faculty of Education, Yamato University, Katayama 2-5-1, Suita, Osaka 564-0082, Japan

Correspondence should be addressed to Maslina Darus; maslina@ukm.edu.my

Received 4 April 2017; Accepted 17 July 2017; Published 20 August 2017