We define two new subclasses, $HS(k, \lambda, b, \alpha)$ and \linebreak $\overline{HS}(k, \lambda, b, \alpha)$, of univalent harmonic mappings using multiplier transformation. We obtain a sufficient condition for harmonic univalent functions to be in $HS(k,\lambda,b,\alpha)$ and we prove that this condition is also necessary for the functions in the class $\overline{HS} (k,\lambda,b,\alpha)$. We also obtain extreme points, distortion bounds, convex combination, radius of convexity and Bernandi-Libera-Livingston integral for the functions in the class $\overline{HS}(k,\lambda,b,\alpha)$.
AbstractThis paper considers the following problem: for what
value r, {r<1} a function that is univalent in the unit
disk {|z|<1} and convex in the disk {|z|<r} becomes starlike
in {|z|<1}. The number r is called the radius of convexity
sufficient for starlikeness in the class of univalent functions.
Several related problems are also considered.
In this paper, we introduce a new comprehensive subclass ΣB(λ,μ,β) of meromorphic bi-univalent functions in the open unit disk U. We also find the upper bounds for the initial Taylor-Maclaurin coefficients |b0|, |b1| and |b2| for functions in this comprehensive subclass. Moreover, we obtain estimates for the general coefficients |bn|(n≧1) for functions in the subclass ΣB(λ,μ,β) by making use of the Faber polynomial expansion method. The results presented in this paper would generalize and improve several recent works on the subject.
Abstract
In this paper we give the upper bounds of the Hankel determinants of the second and third order for the class 𝓢 of univalent functions in the unit disc.
AbstractLet f be analytic in the unit disk $${\mathbb {D}}=\{z\in {\mathbb {C}}:|z|<1 \}$$
D
=
{
z
∈
C
:
|
z
|
<
1
}
, and $${{\mathcal {S}}}$$
S
be the subclass of normalized univalent functions given by $$f(z)=z+\sum _{n=2}^{\infty }a_n z^n$$
f
(
z
)
=
z
+
∑
n
=
2
∞
a
n
z
n
for $$z\in {\mathbb {D}}$$
z
∈
D
. We give sharp bounds for the modulus of the second Hankel determinant $$ H_2(2)(f)=a_2a_4-a_3^2$$
H
2
(
2
)
(
f
)
=
a
2
a
4
-
a
3
2
for the subclass $$ {\mathcal F_{O}}(\lambda ,\beta )$$
F
O
(
λ
,
β
)
of strongly Ozaki close-to-convex functions, where $$1/2\le \lambda \le 1$$
1
/
2
≤
λ
≤
1
, and $$0<\beta \le 1$$
0
<
β
≤
1
. Sharp bounds are also given for $$|H_2(2)(f^{-1})|$$
|
H
2
(
2
)
(
f
-
1
)
|
, where $$f^{-1}$$
f
-
1
is the inverse function of f. The results settle an invariance property of $$|H_2(2)(f)|$$
|
H
2
(
2
)
(
f
)
|
and $$|H_2(2)(f^{-1})|$$
|
H
2
(
2
)
(
f
-
1
)
|
for strongly convex functions.
Abstract
In recent years, the problem of estimating Hankel determinants has attracted the attention of many mathematicians. Their research have been focused mainly on deriving the bounds of
H
2
,
2
{H_{2,2}}
or
H
3
,
1
{H_{3,1}}
over different subclasses of
𝒮
{\mathcal{S}}
. Only in a few papers third Hankel determinants for non-univalent functions were considered. In this paper, we consider two classes of analytic functions with real coefficients. The first one is the class
𝒯
{\mathcal{T}}
of typically real functions. The second object of our interest is
𝒦
ℝ
(
i
)
{\mathcal{K}_{\mathbb{R}}(i)}
, the class of functions with real coefficients which are convex in the direction of the imaginary axis. In both classes, we find lower and upper bounds of the third Hankel determinant. The results are sharp.
Coefficients of a Comprehensive Subclass of Meromorphic Bi-Univalent Functions Associated with the Faber Polynomial Expansion
