• Kyungpook Mathematical Journal
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• v.55 no.2
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• pp.429-438
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• 2015

The objective of this paper is to obtain sharp upper bound for the second Hankel functional associated with the $k^{th}$ root transform $[f(z^k)]^{\frac{1}{k}}$ of normalized analytic function f(z) when it belongs to the class of starlike and convex functions with respect to symmetric points, defined on the open unit disc in the complex plane, using Toeplitz determinants.

• Kyungpook Mathematical Journal
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• v.55 no.3
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• pp.705-714
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• 2015

In this paper, we introduce and investigate an interesting subclass $M_{\Sigma}({\gamma},{\lambda},{\delta},{\varphi})$ of analytic and bi-univalent functions of complex order in the open unit disk ${\mathbb{U}}$. For functions belonging to the class $M_{\Sigma}({\gamma},{\lambda},{\delta},{\varphi})$ we investigate the coefficient estimates on the first two Taylor-Maclaurin coefficients ${\mid}{\alpha}_2{\mid}$ and ${\mid}{\alpha}_3{\mid}$. The results presented in this paper would generalize and improve some recent works of [1],[5],[9].

• Honam Mathematical Journal
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• v.38 no.4
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• pp.701-724
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• 2016

For $f_i$ belonging to various subclasses of univalent functions, we investigate the product given by $h(z)=z{\prod_{i=1}^{n}}(f_i(z)/z)^{{\gamma}_i}$.The largest radius ${\rho}$ is determined such that $h({\rho}z)/{\rho}$ is starlike of order ${\beta}$, $0{\leq}{\beta}$ < 1 or to belong to other subclasses of univalent functions. We also determine the sharp radius of starlikeness of order ${\beta}$and other radius for the convolution f*g of two starlike functions f, g.

• Bulletin of the Korean Mathematical Society
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• v.57 no.6
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• pp.1409-1426
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• 2020

Let 𝒜_n be the class of analytic functions f(z) of the form f(z) = z + ∑^∞_k=n+1 α_kz^k, n ∈ ℕ defined on the open unit disk 𝔻, and let $${\Omega}_n:=\{f{\in}{\mathcal{A}}_n:\|zf^{\prime}(z)-f(z)\|<{\frac{1}{2}},\;z{\in}{\mathbb{D}}\}$$. In this paper, we make use of differential subordination technique to obtain sufficient conditions for the class Ω_n. Writing Ω := Ω₁, we obtain inclusion properties of Ω with respect to functions which map 𝔻 onto certain parabolic regions and as a consequence, establish a relation connecting the parabolic starlike class 𝒮_P and the uniformly starlike UST. Various radius problems for the class Ω are considered and the sharpness of the radii estimates is obtained analytically besides graphical illustrations.

• Journal of the Korean Mathematical Society
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• v.58 no.5
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• pp.1147-1180
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• 2021

By considering the polynomial function 𝜙_car(z) = 1 + z + z²/2, we define the class 𝓢^*_car consisting of normalized analytic functions f such that zf'/f is subordinate to 𝜙_car in the unit disk. The inclusion relations and various radii constants associated with the class 𝓢^*_car and its connection with several well-known subclasses of starlike functions is established. As an application, the obtained results are applied to derive the properties of the partial sums and convolution.

• Communications of the Korean Mathematical Society
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• v.34 no.2
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• pp.451-464
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• 2019

In the present paper, we find the sharp bound for the fourth coefficient of starlike functions f which are normalized by f(0) = 0 = f'(0) - 1 and satisfy the following two-sided inequality: $$1+{\frac{{\gamma}-{\pi}}{2\;{\sin}\;{\gamma}}}\;<\;{\Re}\{{\frac{zf^{\prime}(z)}{f(z)}}\}\;<\;1+{\frac{{\gamma}}{2\;{\sin}\;{\gamma}}},\;z{\in}{\mathbb{D}}$$, where ${\mathbb{D}}:=\{z{\in}{\mathbb{C}}:{\left|z\right|}<1\}$ is the unit disk and ${\gamma}$ is a real number such that ${\pi}/2{\leq}{\gamma}<{\pi}$. Moreover, the sharp bound for the fifth coefficient of f defined above with ${\gamma}$ in a subset of [${\pi}/2,{\pi}$) also will be found.

• Communications of the Korean Mathematical Society
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• v.37 no.2
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• pp.455-472
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• 2022

For j = 0, 1, 2,…, k - 1; k ≥ 2; and - 1 ≤ B < A ≤ 1, we have introduced the functions classes denoted by ST_[j,k](A, B) and K_[j,k](A, B), respectively, called the generalized (j, k)-symmetric starlike and convex functions. We first proved the sharp bounds on |f(z)| and |f'(z)|. Various radii related problems, such as radius of (j, k)-symmetric starlikeness, convexity, strongly starlikeness and parabolic starlikeness are determined. The quantity |a²₃ - a₅|, which provide the initial bound on Zalcman functional is obtained for the functions in the family ST_[j,k]. Furthermore, the sharp pre-Schwarzian norm is also established for the case when f is a member of K_[j,k](α) for all 0 ≤ α < 1.

• Communications of the Korean Mathematical Society
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• v.37 no.4
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• pp.1025-1039
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• 2022

For given non-negative real numbers 𝛼_k with ∑^m_k=1 𝛼_k = 1 and normalized analytic functions f_k, k = 1, …, m, defined on the open unit disc, let the functions F and Fn be defined by F(z) := ∑^m_k=1 𝛼_kf_k(z), and F_n(z) := n^-1 ∑^n-1_j=0 e^-2j𝜋i/nF(e^2j𝜋i/nz). This paper studies the functions f_k satisfying the subordination zf'_k(z)/F_n(z) ≺ h(z), where the function h is a convex univalent function with positive real part. We also consider the analogues of the classes of starlike functions with respect to symmetric, conjugate, and symmetric conjugate points. Inclusion and convolution results are proved for these and related classes. Our classes generalize several well-known classes and the connections with the previous works are indicated.

• Bulletin of the Korean Mathematical Society
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• v.37 no.4
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• pp.659-668
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• 2000

Let's $S_n(p,\alpha)(p,n\in\ N={1,2,3,\cdots},0\leq\ \alpha<1)$ denote tje c;ass pf fimctopms $f(z)=z^p+a_{p+z}z^{p+n}+\cdots$ which are $\rho$-valently starlike of order $\alpha$ in the unit disk.Some criteria for a function f(z) to be in the class $S_n(\rho,\alpha)$ are given.

• Honam Mathematical Journal
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• v.18 no.1
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• pp.59-77
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• 1996

In this paper, we study how the second coefficient in the power series expansion of functions in $P_n$(a; b, M), $F_n$(a; b, M) and $G_n$(a; b, M) influences certain properties like distortion, ${\gamma}-spiral$ radius and radius of starlikeness of these classes.