Open Math. 2017; 15: 331–339
Open Mathematics
Open Access
Research Article
Nak Eun Cho, Naveen Kumar Jain*, and V. Ravichandran
Convex combination of analytic functions
DOI 10.1515/math-2017-0021
Received August 18, 2016; accepted January 26, 2017.
Abstract: Radii of convexity, starlikeness, lemniscate starlikeness and close-to-convexity are determined for the
convex combination of the identity map and a normalized convex function F given by f .z/ D ˛z C .1
˛/F .z/.
Keywords: Convex function, Starlike function, Close-to-convex function, Lemniscate of Bernoulli, Radius of
starlikeness, Convolution, Convex combination
MSC: 30C80, 30C45
1 Introduction
Let A be the class of analytic functions f defined on the unit disc D D fz 2 C W jzj < 1g, and normalized by
f .0/ D 0 D f 0 .0/ 1. Let S ; ST ; CV and CCV denote the subclasses of A consisting of functions univalent,
starlike, convex and close-to-convex respectively. Recall that a function f 2 A is close-to-convex if there exists a
convex function g such that Re .f 0 .z/=g 0 .z// > 0 for all z 2 D. The class ST .ˇ/ of starlike functions of order
ˇ, 0 ˇ < 1, consists of f 2 A satisfying Re .zf 0 .z/=f .z// > ˇ for all z 2 D and ST WD ST .0/. The class
CV .ˇ/ of convex functions of order ˇ is defined by CV .ˇ/ D ff 2 A W zf 0 .z/ 2 ST .ˇ/g and CV WD CV .0/. The
class SL of lemniscate starlike functions, introduced by Sokół and Stankiewicz [1], consists of f 2 A satisfying
j .zf 0 .z/=f .z//2 1j < 1 for all z 2 D, or, equivalently, if zf 0 .z/=f .z/ lies in the region bounded by the right-half
of the lemniscate of Bernoulli given by jw 2 1j < 1. For recent investigation on the class SL, see [1–5]. Another
class of our interest is the class Mˇ , ˇ > 1, consisting of f 2 A satisfying Re .zf 0 .z/=f .z// < ˇ for all z 2 D.
The class Mˇ was investigated by Uralegaddi et al. [6], while its subclass was investigated by Owa and Srivastava
[7]. Related radius problem for this class can be found in [8] and [9].
Properties of linear combination, in particular, convex combination of functions belonging to various classes
of functions were initially investigated by Rahmanov in 1952 and 1953 [10, 11]. The survey article of Campbell
[12] provides several results concerning various combination of univalent functions as well as of locally univalent
functions. Convex combination of univalent functions and the identity function were investigated by several authors
(see Merkes [13] and references therein as well as [14]); in particular, Merkes [13] proved some results related to
the present investigation. Obradovic and Nunokowa [15] investigated functions f 2 A satisfying the following
condition
zf 00 .z/
>0
.z 2 D/
(1)
Re 1 C 0
f .z/ ˛
for some ˛ 2 Œ 1; 1/ and obtained the following result.
Nak Eun Cho: Department of Applied Mathematics, Pukyong National University, Busan 608-737, South Korea,
E-mail: [email protected]
*Corresponding Author: Naveen Kumar Jain: Department of Mathematics, University of Delhi, Delhi-110007, India,
E-mail: [email protected]
V. Ravichandran: Department of Mathematics, University of Delhi, Delhi-110007, India,
E-mail: [email protected]; [email protected]
© 2017 Cho et al., published by De Gruyter Open.
This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivs 3.0 License.
332
N.E. Cho et al.
Theorem 1.1 ([15, Theorem 2]). If f 2 A satisfies the condition (1), then (i) f 2 ST for
f 2 CCV for 31 ˛ < 1.
1
3
˛
1
,
3
and (ii)
If the function f is the convex combination f .z/ D ˛z C .1 ˛/F .z/, then the condition (1) is equivalent to the
conditions that F 2 CV . If two subclasses G and F of A are given, the G -radius of F , denoted by RG .F /, is the
largest number R such that f .rz/=r 2 G for 0 < r < R, and for all f 2 F . Whenever G is characterized by
possesing a geometric property P , the number R is also referred to as the radius of property P for the class F . In
this paper, we investigate radius problem for functions f satisfying the condition (1) to belong to one of the classes
introduced above. We also prove the correct results corresponding to [15, Theorem 1(a) and Theorem 2(a), p. 100]
p
that f 2 CV if f 2 A satisfies the condition (1) for some 0 ˛ .12 2 15/=9. Their result is correct only
when ˛ D 0. Unlike the radii problems associated with starlikeness and convexity, where a central feature is the
estimate for the real part of the expressions zf 0 .z/=f .z/ or 1 C zf 00 .z/=f 0 .z/ respectively, the SL-radius problems
are tackled by first finding the disc that contains the values of zf 0 .z/=f .z/ or 1 C zf 00 .z/=f 0 .z/. The techniques
used in this paper are earlier used for the class of uniformly convex functions investigated in [16–26].
P
P
n
n
For two analytic functions f; g 2 A with f .z/ D z C 1
D zC 1
nD2 an z and g.z/P
nD2 bn z , their
1
convolution or Hadamard product, denoted by f g, is defined by .f g/.z/ WD z C nD2 an bn z n . We need the
following results.
Theorem 1.2 ([28, Theorem 2.1-2.2, p. 125]). The classes ST , CV and CCV are closed under convolution with
functions in CV .
Theorem 1.3. The classes ST .ˇ/, SL and M.ˇ/
T
ST are closed under convolution with functions in CV .
p
Lemma 1.4 ([4, Lemma 2.2, p. 6559]). Let 0 < a < 2. If ra is given by
8p
p
< 1 a2 .1 a2 / 1=2
.0 < a 2 2=3/
ra D p
p
p
: 2 a
.2 2=3 a < 2/;
then fw W jw
aj < ra g fw W jw 2
1j < 1g.
Theorem 1.3 is a special case of results of Shanmugam [27, Theorem 3.3, p. 336; Theorem 3.5, p. 337] (see also Ma
and Minda [18, Theorem 5, p. 167]).
2 Radii problems associated with convex combinations
For functions satisfying the condition (1), we determine, in the first part of the following theorem, the range of ˛
so that the function is starlike of order ˇ while the other parts of the theorem provide the radius of starlikeness of
order ˇ.
Theorem 2.1. Let 1 ˛ < 1, 0 ˇ < 1 and f 2 A satisfy the condition (1).
(a) If 0 ˇ 1=2 and j˛j .1 2ˇ/=.3 2ˇ/, then f 2 ST .ˇ/:
(b) If either
(i) 0 ˇ 1=2, and .1 2ˇ/=.3 2ˇ/ < ˛ < 1, or
(ii) 1=2 < ˇ < 1, and . ˇ C 2ˇ 2 /=.6 7ˇ C 2ˇ 2 / < ˛ < 1,
then f .1 z/=.1 / 2 ST .ˇ/ where 1 D 1 .˛; ˇ/ is given by
1 12
0
B
B
1 .˛; ˇ/ D B p
@ 4 .1
˛.3
˛/˛ 2 .˛.ˇ
2/
2ˇ/2
ˇ/.ˇ
.1
2ˇ/2
2
1/ C ˛ 1 C 4ˇ 4ˇ
C ˛ 2 7 12ˇ C 4ˇ 2
C
C
C :
A
Convex combination of analytic functions
333
(c) If either
(i) 0 ˇ 1=2, and 1 ˛ < .1 2ˇ/=. 3 C 2ˇ/, or
(ii) 1=2 < ˇ < 1, and 1 ˛ < . ˇ C 2ˇ 2 /=.6 7ˇ C 2ˇ 2 /,
then f .0 z/=0 2 ST .ˇ/ where 0 D 0 .˛; ˇ/ is given by
0 .˛; ˇ/ D
2.1
2/ C ˇ C
˛.ˇ
p
ˇ/
ˇ/2 C .˛.ˇ
4˛.1
2/ C ˇ/2
:
Proof. Define the function g W D ! C by
˛z 2
:
1 z
For a fixed ˛ 2 Œ 1; 1/, define the function f1 W D ! C by
g.z/ D
f1 .z/ D
z
(2)
f .z/ ˛z
:
1 ˛
(3)
If f satisfies the condition (1), then it follows that the function f1 2 CV . With the function g defined by (2), the
equation (3) shows that
f .z/ D ˛z C .1 ˛/f1 .z/ D f1 .z/ g.z/:
If g is starlike in the disc D , then g.z/= is starlike and hence, by Theorem 1.2, f1 .z/ .g.z/=/ is starlike or
equivalently f1 g is starlike in the disc D . In view of this, it is enough to investigate the radius of starlikeness of
the function g given by (2).
For the function g given by (2), we have
zg 0 .z/
1
D
g.z/
1 z
˛z
:
˛z
1
With z D re it and x D cos t , a calculation shows that
0 1 r cos t
zg .z/
r˛.r˛
D
Re
C
2
g.z/
1Cr
2r cos t
1 C r 2 ˛2
D
cos t /
2r˛ cos t
1 C 2r 2 ˛ 2 C r 4 ˛ 2 r.1 C 3˛/.1 C r 2 ˛/x C 4r 2 ˛x 2
:
.1 C r 2 2rx/.1 2rx˛ C r 2 ˛ 2 /
Therefore, g is starlike of order ˇ in jzj < 1 if, for all 0 r < 1 and for all x 2 Œ 1; 1, we have
1 C 2r 2 ˛ 2 C r 4 ˛ 2
ˇ.1 C r 2
r.1 C 3˛/.1 C r 2 ˛/x C 4r 2 ˛x 2
2rx/.1
2rx˛ C r 2 ˛ 2 / 0:
This inequality is equivalent to
h.r; x/ W D 4r 2 ˛.1
ˇ/x 2 C r.1 C r 2 ˛/. 1
2 2
4 2
C 1 C 2r ˛ C r ˛
ˇ
3˛ C 2ˇ C 2˛ˇ/x
2
r 2 ˛2 ˇ
r ˇ
r 4 ˛2 ˇ 0
for 0 r < 1 and for all x 2 Œ 1; 1. It follows that the derivative of function h.r; x/ with respect to x vanishes for
x D x0 D
.1 C r 2 ˛/.1 2ˇ C ˛.3
8r˛.1 ˇ/
2ˇ//
.˛ ¤ 0/:
It should be noted for later use that, for ˇ 1=2 and ˛ 0,
h.r; 1/
h.r; 1/ D .1
r/.1
r˛/.1 C r 2 ˛.1
.1 C r/.1 C r˛/.1 C r 2 ˛.1
2
D 2r.1 C r ˛/. 1
2
ˇ/
ˇ C r.˛.ˇ
ˇ/
3˛ C 2ˇ C 2˛ˇ/
2r.1 C r ˛/. 2˛/ 0;
ˇ C r.˛.2
2/ C ˇ//
ˇ/
ˇ//
(4)
334
N.E. Cho et al.
and so we have h.r; 1/ h.r; 1/.
(a) Case (i) Let 0 ˇ 1=2 and 0 ˛ .1 2ˇ/=.3 2ˇ/. If ˇ D 1=2, then ˛ D 0, h.r; x/ D .1
and hence minjxj1 h.r; x/ > 0 for 0 r < 1. If 0 ˇ < 1=2 and ˛ D 0, then
h.r; x/ D 1
r 2 ˇ C .2ˇ
ˇ
r 2 /=2
1/rx
and so
min h.r; x/ D h.r; 1/ D .1
jxj1
ˇ C ˇr/ > 0
r/.1
for 0 r < 1:
If 0 ˇ < 1=2 and 0 < ˛ .1 2ˇ/=.3 2ˇ/, then it can be verified that x0 > 1. In view of this and from (4), it
follows that minjxj1 h.r; x/ D h.r; 1/: Since h.r; 1/ is a decreasing function of ˛, we have, for 0 r < 1,
.1
min h.r; x/ D h.r; 1/ jxj1
r/.3 2ˇ C r. 1 C 2ˇ//
3 5ˇ C 2ˇ 2 C r 2 C 8ˇ 4ˇ 2 C r 2 1
.3 2ˇ/2
3ˇ C 2ˇ 2
> 0:
Therefore, for 0 r < 1, minjxj1 h.r; x/ > 0 and so g is starlike of order ˇ.
Case (ii) Let 0 ˇ < 1=2 and .1 2ˇ/=. 3 C 2ˇ/ < ˛ < 0. For fixed r, the second partial derivative of h.r; x/
with respect to x is negative and so the minimum of h.r; x/ in Œ 1; 1 is attained at the end points x D ˙1. Using (4)
and the fact that h.r; 1/ is a decreasing function of ˛, minjxj1 h.r; x/ > 0 for 0 r < 1, and minjxj1 h.r; x/ > 0.
Therefore, g is starlike of order ˇ.
(b) Case (i) Let 0 ˇ 1=2 and .1 2ˇ/=.3 2ˇ/ < ˛ < 1. It can be seen that 1 x0 1 if
p
.1 ˛/˛.˛.3 2ˇ/2 .1 2ˇ/2 /
4˛.1 ˇ/
r
WD 0 :
˛.1 2ˇ C ˛.3 2ˇ//
So, for 0 r < 1 , minjxj1 h.r; x/ D h.r; x0 / > 0, where
h.r; x0 / D
1 ˛
.9˛
16˛.1 ˇ/
1 C 4ˇ
4ˇ 2 C 4˛ˇ 2 /˛r 2 C .9˛
4ˇ 2 C 4˛ˇ 2
12˛ˇ
1 C 4ˇ
12˛ˇ
2.1 C 7˛ C 4ˇ
12˛ˇ
4ˇ 2 C 4˛ˇ 2 /˛ 2 r 4 /:
Notice that the number 1 .˛; ˇ/ is the root of h.r; x0 / D 0. For 0 r < 0 , since
s.˛/ D 1 C r 2 ˛.1
ˇ/
ˇ C r.˛.ˇ
2/ C ˇ/
is a decreasing function of ˛,
min h.r; x/ D h.r; 1/ D .1
jxj1
r/.1
r˛/.1 C r 2 ˛.1
ˇ/
ˇ C r.˛.ˇ
2/ C ˇ//
(5)
.1
r/4 .1
ˇ/ > 0:
Therefore, minjxj1 h.r; x/ 0 for 0 r < 1 and hence g is starlike of order ˇ in jzj < 1 .
Case (ii) Let 1=2 < ˇ < 1 and .1 2ˇ/=. 3 C 2ˇ/ < ˛ < 1. Let r < 1 . It can be shown that
p
.1 ˛/˛.˛.3 2ˇ/2 .1 2ˇ/2 /
4˛.1 ˇ/
WD 1 :
1 x0 1 for r ˛.1 2ˇ C ˛.3 2ˇ//
This shows that for 1 r < 1 , minjxj1 h.r; x/ D h.r; x0 / > 0.
On the other hand if 0 < r < 1 , then x0 > 1. Now proceeding as in case (i) we get minjxj1 h.r; x/ D
h.r; 1/ > 0. Thus g is starlike of order ˇ in jzj < 1 .
Case (iii) Let 1=2 < ˇ < 1 and . ˇC2ˇ 2 /=.6 7ˇC2ˇ 2 / < ˛ .1 2ˇ/=. 3C2ˇ/. Let r2 be the root of the
equation x0 D 1. Then r2 1 . Let r < 1 . If r r2 then 1 x0 1. Thus minjxj1 h.r; x/ D h.r; x0 / > 0.
On the other hand if r < r2 , then x0 < 1. Also 1 < 0 . Thus minjxj1 h.r; x/ D h.r; 1/ > 0 in jzj < 1 and
hence g is starlike of order ˇ in jzj < 1 .
(c) Case (i) Let 0 ˇ 1=2 and 1 ˛ < .1 2ˇ/=. 3 C 2ˇ/. For fixed r, the second derivative
test shows that the minimum of h.r; x/ in Œ 1; 1 is attained at the end points x D ˙1. It follows from (4) that
minjxj1 h.r; x/ D h.r; 1/. Notice that
h.r; 1/ D .1 C r/.1 C r˛/.1
ˇ
r.ˇ C ˛. 2 C r. 1 C ˇ/ C ˇ///
Convex combination of analytic functions
335
and the number 0 .˛; ˇ/ is the root of h.r; 1/ D 0. Since h.r; 1/ > 0 for r < 0 , minjxj1 h.r; x/ > 0 in
jzj < 0 . Thus g is starlike of order ˇ in jzj < 0 .
Case (ii) Let 1=2 < ˇ < 1 and 0 < ˛ . ˇ C 2ˇ 2 /=.6 7ˇ C 2ˇ 2 /. Then r2 0 . Let r < 0 . Since
x0 < 1, we have minjxj1 h.r; x/ D h.r; 1/ > 0 in jzj < 0 . Thus g is starlike of order ˇ in jzj < 0 .
Case (iii) Let 1=2 < ˇ < 1 and 1 ˛ 0. For fixed r, the minimum of h.r; x/ in Œ 1; 1 is attained at the
end points x D ˙1. Also h.r; 1/ h.r; 1/ > 0. Thus minjxj1 h.r; x/ D h.r; 1/ > 0 in jzj < 0 . Thus g is
starlike of order ˇ in jzj < 0 .
To prove the sharpness, consider the functions f and g given by
f .z/ D g.z/ D
z
˛z 2
:
1 z
(6)
For the function f given in (6), we have
zf 00 .z/
1Cz
Re 1 C 0
D Re
> 0:
f .z/ ˛
1 z
The function f satisfies the condition (1) and since radius is sharp for the function g, the sharpness follows.
Remark 2.2. If 1=3 ˛ 1=3 and f 2 A satisfy the condition (1), then f 2 ST .ˇ/ where
ˇD
1 3j˛j
0:
2.1 j˛j/
In particular, we have the following corollary.
Corollary 2.3 ([15, Theorem 2(b), p. 100]). If f satisfies the condition (1) for some ˛ with j˛j 13 , then f 2 ST .
For other ranges of ˛, we have the following corollary.
Corollary 2.4. The radius of starlikeness of the class of functions f 2 A satisfying the condition (1) for ˛ 2
. 1; 1=3 [ Œ1=3; 1/ is given by
8
12
ˆ
<
9˛ p
1
if 13 ˛ < 1;
p
˛C7˛ 2 C4 2 .1 ˛/˛ 3
rD
ˆ
:p 1
if 1 < ˛ 31 :
˛.˛ 1/ ˛
Theorem 2.5. For 1 ˛ < 1, the SL-radius of the class of functions f 2 A satisfying the condition (1) is given
by
p
2
2
2 .˛/ D
:
r
p
p 1C˛
2˛ C .1 ˛/ 1 3˛ C 2 2˛
Proof. First we observe that
j.1
˛z/.1
z/j .1
˛r/.1
r/;
jzj D r < 1:
(7)
For 0 ˛ < 1, (7) is trivial. For 1 ˛ < 0, the inequality (7) holds as the function
h.r; x/ WD j.1
˛z/.1
z/j2 D .1
2r˛x C r 2 ˛ 2 /.1
2rx C r 2 /;
where x D cos t, is a decreasing function. Since the function g given by (2) satisfies
ˇ 0
ˇ ˇ
ˇ
ˇ zg .z/
ˇ ˇ
ˇ
.1 ˛/z
r.1 ˛/
ˇ
ˇDˇ
ˇ
1
ˇ g.z/
ˇ ˇ .1 z/.1 ˛z/ ˇ .1 r/.1 ˛r/ ;
Equation 8 and Lemma 1.4 yield
ˇ
ˇ zg 0 .z/ 2
ˇ
ˇ
ˇ g.z/
ˇ
ˇ
ˇ
1ˇ < 1
ˇ
.jzj < r/
(8)
336
N.E. Cho et al.
provided
1
p
2C
p 2 1
p
2˛ C ˛ r C ˛ 1
p 2
2 r 0
or r 2 :
Thus g.2 z/=2 2 SL. As the function f satisfies (1), the function f1 defined by (3) is convex. Hence, by Theorem
1.3, we have
f .2 z/
g.2 z/
D f1 .z/ 2 SL
2
2
or, equivalently
ˇ
ˇ
ˇ
1ˇ < 1
ˇ
ˇ
ˇ zf 0 .z/ 2
ˇ
ˇ
ˇ f .z/
For z D 2 , the function g given by (2) satisfies
ˇ ˇ
ˇ
2
ˇ ˇ
ˇ zg 0 .z/ 2
˛2
1
ˇ ˇ
ˇ
1ˇ D ˇ
ˇ
ˇ ˇ 1 2 1 ˛2
ˇ g.z/
.jzj < 2 /:
ˇ ˇ
2
ˇ ˇ
2 .1 ˛/
ˇ ˇ
1ˇ D ˇ
C1
ˇ ˇ .1 2 /.1 ˛2 /
ˇ
ˇ
ˇ
1ˇ D 1:
ˇ
Thus, the result is sharp for the function
f .z/ D g.z/ D
z
˛z 2
:
1 z
p
2.
Corollary 2.6. The SL-radius of the class CV of convex functions is 2
Theorem 2.7. For 1=3 ˛ 1=3 and ˇ > 1, the Mˇ \ ST radius of the class of functions f 2 A satisfying
the condition (1) is given by
2.ˇ
3 .˛; ˇ/ D
2˛ C ˛ˇ C
ˇ
q
.˛
1/
1/ 4˛
4˛ˇ
ˇ 2 C ˛ˇ 2
:
Proof. From the inequality (8) for the function g given by (2), we have
Re
r.1 ˛/
zg 0 .z/
1C
ˇ
g.z/
.1 r/.1 ˛r/
provided
1
ˇ C . 2˛ C ˇ C ˛ˇ/r C ˛.1
ˇ/r 2 0:
The last inequality holds if 0 < r 3 .˛; ˇ/. Thus g.3 z/=3 2 Mˇ . Since the function f satisfies (1), the
function f1 defined by (3) is convex. Hence, by Theorem 1.3, we have
f .3 z/
g.3 z/
D f1 .z/ 2 Mˇ
3
3
or
Re
zf 0 .z/
f .z/
Also, by Corollary 2.3,
0 < Re
Thus f .3 z/=3 2 Mˇ
T
<ˇ
zf 0 .z/
f .z/
.jzj < 3 /:
.jzj < 1/:
ST : For z D 3 , the function g given by (2) satisfies
Re
zg 0 .z/
3 .1 ˛/
D1C
D ˇ:
g.z/
.1 3 /.1 ˛3 /
Thus the result is sharp for the function
f .z/ D g.z/ D
Corollary 2.8. For ˇ > 1, the Mˇ
T
z
˛z 2
:
1 z
ST -radius of the class CV of convex functions is 1
ˇ
1
.
Convex combination of analytic functions
337
The results [15, Theorem 1(a) and Theorem 2(a), p. 100] of Obradovic and Nunokawa are incorrect. The correct
version of these results are given in the following theorem.
Theorem 2.9. For 1 ˛ < 1, the radius of convexity of the class of functions f 2 A satisfying the condition (1)
is
(
r1 ; if 0 ˛ < 1;
4 .˛/ D
(9)
r2 if
1 ˛ 0;
where r1 2 .0; 1 is the root of the equation in r:
1
r 2 .1 C ˛/
r 4 ˛.1
2˛/
q
2r 3 ˛ .1
r 2˛ D 0
˛/ 1
and r2 2 .0; 1 is the root of the equation in r:
1 C 6˛r
1 8˛ 8˛ 2 r 2 C 2˛ .1 C 9˛/ r 3 C 15˛ 2 r 4 C 6˛ 2 r 5 C ˛ 2 r 6 D 0:
Proof. If the function f satisfies (1) with ˛ D 0, then the function f is convex and so 4 .0/ D 1 as claimed. Now,
assume that ˛ ¤ 0. For the function g given by (2), a calculation shows that, with x D cos t ,
zg 00 .z/
1 r2
Re 1 C 0
D
g .z/
1 C r 2 2r cos t
2r˛. .1 C 3r 2 ˛/ cos t C r..2 C r 2 /˛ C cos.2t ///
1 C r 2 .4 C r 2 /˛ 2 C 2r˛. 2.1 C r 2 ˛/ cos t C r cos.2t //
.r; x/
D
h.r; x/
C
where
h.r; x/ D 1 C r 2
2rx
1 C r 2 4 C r 2 ˛2
2r˛ r C 2x
2rx 2 C 2r 2 x˛
6r˛ 1
r 2 C 3r 2 ˛ C r 4 ˛ x
and
4r 2 ˛ C 8r 2 ˛ 2 C 3r 4 ˛ 2 C r 6 ˛ 2
C 12r 2 ˛ 1 C r 2 ˛ x 2 8r 3 ˛x 3 :
.r; x/ D 1
r2
(10)
Then
@
.r; x/ D
@x
A calculation shows that
6r˛ 1
@
.r; x/ D 0
@x
and
r 2 C 4r 2 x 2 C 3r 2 ˛ C r 4 ˛ C 24r 2 x˛ 1 C r 2 ˛ :
if x D x0 D
q
@2
3
.r;
x
/
D
24r
˛
.1
0
@x 2
1 C r 2˛
˛/ 1
r
q
.1
˛/ 1
r 2˛
2r
r 2 ˛ > 0 for 0 < ˛ < 1:
So for 0 < ˛ < 1,
˛/ 1 r 2 ˛ 1 r 2 .1 C ˛/
q
2r 3 ˛ .1 ˛/ 1 r 2 ˛ > 0:
min .r; x/ D .r; x0 / D .1
jxj1
r 4 ˛.1
2˛/
On the other hand, if 1 < ˛ < 0, then, for r < r2 , it can be verified that
min .r; x/ D .r; 1/ D 1 C 6˛r
1 8˛ 8˛ 2 r 2 C 2˛ .1 C 9˛/ r 3
jxj1
C 15˛ 2 r 4 C 6˛ 2 r 5 C ˛ 2 r 6 > 0:
338
N.E. Cho et al.
Thus g.4 z/=4 2 CV . Since the function f satisfies (1), the function f1 defined by (3) is convex. Hence
f .4 z/
g.4 z/
D f1 .z/ 2 CV
4
4
or
zf 00 .z/
Re 1 C 0
>0
f .z/
.jzj < 4 /:
The result is sharp for the function f given by (6).
Theorem 2.10. For 1 ˛ <
condition (1) is given by
1=3, the radius of close-to-convexity of the class of functions f 2 A satisfying the
5 .˛/ D p
1
˛.˛
1/
˛
:
Proof. The function g1 W D ! C by
g1 .z/ D
ln.1
z/
.z 2 D/
is clearly convex in D. For the functions g given by (2) and g1 above, we have
!
1 C ˛r 2 .2˛ C 1 C ˛r 2 /rx C 2˛r 2 x 2
g 0 .re i t /
D
Re
1 C r 2 2rx
g10 .re i t /
(11)
where x WD cos t. Let h W Œ 1; 1 ! R be defined by
h.r; x/ D 1 C ˛r 2
.2˛ C 1 C ˛r 2 /rx C 2˛r 2 x 2 :
Then
@
h.r; x/ D 0
@x
if x D x0 D
1 C 2˛ C r 2 ˛
4r˛
and
@2
h.r; x/ D 4r 2 ˛ < 0:
@x 2
Therefore, for a fixed r, the minimum of h.r; x/ is attained at x D ˙1. Since
h.r; 1/ D 2r.1 C 2˛ C r 2 ˛/ < 0;
h.r; 1/
it follows that
min h.r; x/ D h.r; 1/ D .1 C r/ 1 C 2r˛ C r 2 ˛ > 0
jxj1
for
r < 5 :
Thus g.5 z/=5 2 CCV . Since the function f1 defined by (3) is convex as f satisfies (1), we have, by Theorem 1.2,
f .5 z/
g.5 z/
D f1 .z/ 2 CCV
5
5
or
Re
g 0 .z/
g10 .z/
>0
.jzj < 5 /:
The result is sharp for the function
f .z/ D g.z/ D
z
˛z 2
:
1 z
Acknowledgement: This research was supported by the Basic Science Research Program (No.
2016R1D1A1A09916450) through the National Research Foundation of Korea (NRF) funded by the Ministry of
Education, Science and Technology. The authors are thankful to the referees for their comments.
Convex combination of analytic functions
339
References
[1]
[2]
[3]
[4]
[5]
[6]
[7]
[8]
[9]
[10]
[11]
[12]
[13]
[14]
[15]
[16]
[17]
[18]
[19]
[20]
[21]
[22]
[23]
[24]
[25]
[26]
[27]
[28]
Sokół J. and Stankiewicz J., Radius of convexity of some subclasses of strongly starlike functions, Zeszyty Nauk. Politech.
Rzeszowskiej Mat., 1996, 19, 101–105.
Ali R.M., Jain N.K. and Ravichandran V., On the largest disc mapped by sum of convex and starlike functions, Abstr. Appl. Anal.,
2013, Art. ID 682413, 12 pp.
Ali R.M., Cho N.E., Jain N. and Ravichandran V., Radii of starlikeness and convexity of functions defined by subordination with
fixed second coefficients, Filomat, 2012, 26(3), 553–561.
Ali R.M., Jain N.K. and Ravichandran V., Radii of starlikeness associated with the lemniscate of Bernoulli and the left-half plane,
Appl. Math. Comput., 2012, 218(11), 6557–6565.
Sokół J., Radius problems in the class SL, Appl. Math. Comput., 2009, 214(2), 569–573.
Uralegaddi B.A., Ganigi M.D. and Sarangi S.M., Univalent functions with positive coefficients, Tamkang J. Math., 1994, 25, 225–
230, no. 3.
Owa S. and Srivastava H.M., Some generalized convolution properties associated with certain subclasses of analytic functions, J.
Inequal. Pure Appl. Math., 2002, 3(3), Article 42, 13 pp.
Ali R.M., Mohd M.H., Keong L.S. and Ravichandran V., Radii of starlikeness, parabolic starlikeness and strong starlikeness for
Janowski starlike functions with complex parameters, Tamsui Oxf. J. Inf. Math. Sci., 2011, 27(3), 253–267.
Ravichandran V., Silverman H., Khan M.H., Subramanian K.G., Radius problems for a class of analytic functions, Demonstratio
Math., 2006, 39(1), 67–74.
Rahmanov B.N., On the theory of univalent functions, Doklady Akad. Nauk SSSR (N.S.), 1952, 82, 341–344.
Rahmanov B.N., On the theory of univalent functions, Doklady Akad. Nauk SSSR (N.S.), 1953, 88, 413–414.
Campbell D.M., A survey of properties of the convex combination of univalent functions, Rocky Mountain J. Math., 1975, 5(4),
475–492.
Merkes E.P., On the convex sum of certain univalent functions and the identity function, Rev. Colombiana Mat., 1987, 21(1), 5–11.
Ali R.M., Subramanian K.G., Ravichandran V. and Ahuja O.P., Neighborhoods of starlike and convex functions associated with
parabola, J. Inequal. Appl., 2008, Art. ID 346279, 9 pp.
Obradovic M. and Nunokawa M., An application of convolution properties to univalence in the unit disc, Math. Japon., 1997, 46(1),
99–102.
Goodman A.W., On uniformly starlike functions, J. Math. Anal. Appl., 1991, 155(2), 364–370.
Goodman A.W., On uniformly convex functions, Ann. Polon. Math., 1991, 56(1), 87–92.
Ma W.C. and Minda D., A unified treatment of some special classes of univalent functions, in Proc. Conf. Complex Analysis
(Tianjin, 1992), 157–169, Int. Press, Cambridge, MA.
Ravichandran V., Rønning F. and Shanmugam T.N., Radius of convexity and radius of starlikeness for some classes of analytic
functions, Complex Variables Theory Appl., 1997, 33(1-4), 265–280.
Ravichandran V. and Shanmugam T.N., Radius problems for analytic functions, Chinese J. Math., 1995, 23(4), 343–351.
Rønning F., On uniform starlikeness and related properties of univalent functions, Complex Variables Theory Appl., 1994, 24(3-4),
233–239.
Rønning F., Some radius results for univalent functions, J. Math. Anal. Appl., 1995, 194(1), 319–327.
Rønning F., Uniformly convex functions and a corresponding class of starlike functions, Proc. Amer. Math. Soc., 1993, 118(1),
189–196.
Rønning F., On starlike functions associated with parabolic regions, Ann. Univ. Mariae Curie-Skłodowska Sect. A, 1991, 45, 117–
122, 1992.
Rønning F., A survey on uniformly convex and uniformly starlike functions, Ann. Univ. Mariae Curie-Skłodowska Sect. A, 1993, 47,
123–134.
Shanmugam T.N. and Ravichandran V., Certain properties of uniformly convex functions, in Computational Methods and Function
Theory 1994 (Penang), 319–324, Ser. Approx. Decompos., 5 World Sci. Publ., River Edge, NJ.
Shanmugam T.N., Convolution and differential subordination, Internat. J. Math. Math. Sci., 1989, 12(2), 333–340.
Ruscheweyh St. and Sheil-Small T., Hadamard products of Schlicht functions and the Pólya-Schoenberg conjecture, Comment.
Math. Helv., 1973, 48, 119–135.