• Journal of applied mathematics & informatics
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• 제36권5_6호
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• pp.567-574
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• 2018

Let $f:f(z)=z+{\sum^{{\infty}}_{n=2}}a_nz^n$ be analytic in the open unit disc E. Then f is said to belong to the class $M_{\alpha}$ of alpha-convex functions, if it satisfies the condition ${\Re}\{(1-{{\alpha})}{\frac{zf^{\prime}(z)}{f(z)}}+{{\alpha}}{\frac{(zf^{\prime}(z))^{\prime})}{f^{\prime}(z)}}\}$ > 0, ($z{\in}E$). In this paper, we introduce and study q-analogue of the class $M_{\alpha}$ by using concepts of Quantum Analysis. It is shown that the functions in this new class $M(q,{\alpha})$ are q-starlike. A problem related to q-Bernardi operator is also investigated.

• 대한수학회보
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• 제51권5호
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• pp.1551-1559
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• 2014

We consider functions that map the open unit disc conformally onto the complement of an unbounded convex set with opening angle ${\pi}{\alpha}$, ${\alpha}{\in}(1,2]$, at infinity. We derive the exact interval for the variability of the real Taylor coefficients of these functions and we prove that the corresponding complex Taylor coefficients of such functions are contained in certain discs lying in the right half plane. In addition, we also determine generalized central functions for the aforesaid class of functions.

• 대한수학회보
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• 제48권1호
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• pp.169-182
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• 2011

The main object of this paper is to introduce and investigate new subclasses of normalized analytic functions in the open unit disc $\mathbb{U}$, which generalize the familiar class of k-starlike functions. The various properties and characteristics for functions belonging to these classes derived here include (for example) coefficient inequalities, distortion theorems involving fractional calculus, extreme points, integral operators and integral means inequalities.

• Kyungpook Mathematical Journal
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• 제55권2호
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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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• 제51권1호
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• pp.77-85
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• 2011

In this paper, we introduce the integral operator $I_{\beta}$($p_1$, ${\ldots}$, $p_n$; ${\alpha}_1$, ${\ldots}$, ${\alpha}_n$)(z) analytic functions with positive real part. The radius of convexity of this integral operator when ${\beta}$ = 1 is determined. In particular, we get the radius of starlikeness and convexity of the analytic functions with Re {f(z)/z} > 0 and Re {f'(z)} > 0. Furthermore, we derive sufficient condition for the integral operator $I_{\beta}$($p_1$, ${\ldots}$, $p_n$; ${\alpha}_1$, ${\ldots}$, ${\alpha}_n$)(z) to be analytic and univalent in the open unit disc, which leads to univalency of the operators $\int\limits_0^z(f(t)/t)^{\alpha}$dt and $\int\limits_0^z(f'(t))^{\alpha}dt$.

• 한국수학교육학회지시리즈B:순수및응용수학
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• 제22권3호
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• pp.263-273
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• 2015

In this paper, a boundary version of Schwarz lemma is investigated. For the function holomorphic f(z) = a + c_pz^p + c_p+₁z^p+1 + ... defined in the unit disc satisfying |f(z) − 1| < 1, where 0 < a < 2, we estimate a module of angular derivative at the boundary point b, f(b) = 2, by taking into account their first nonzero two Maclaurin coefficients. The sharpness of these estimates is also proved.

• Kyungpook Mathematical Journal
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• 제56권3호
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• pp.867-876
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• 2016

The object of the present paper is to investigate the starlikeness of the class of functions $f(z)=z^p+{\sum\limits_{k=n}^{\infty}}a_p+k^{z^{p^{+k}}} (p,n{\in}{\mathbb{N}}=\{1,2,{\ldots}\})$ which are analytic and p-valent in the unit disc U and satisfy the condition $\|(1-{\lambda}({\frac{f(z)}{z^p}})^{\alpha}+{\lambda}{\frac{zf^{\prime}(z)}{pf(z)}}({\frac{f(z)}{z^p}})^{\alpha}-1\|$ < ${\mu}$ (0 < ${\mu}{\leq}1$, ${\lambda}{\geq}0$, ${\alpha}$ > 0, $z{\in}U$). The starlikeness of certain integral operator are also discussed. The results obtained generalize the related works of some authors and some other new results are also obtained.

• Kyungpook Mathematical Journal
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• 제60권3호
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• pp.519-534
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• 2020

Let H(𝔻) be the space of all holomorphic functions on the open unit disc 𝔻 in the complex plane ℂ. In this paper, we investigate the boundedness and compactness of the generalized integration operator $$I^{(n)}_{g,{\varphi}}(f)(z)=\normalsize\displaystyle\smashmargin{2}{\int\nolimits_0}^z\;f^{(n)}({\varphi}({\xi}))g({\xi})\;d{\xi},\;z{\in}{\mathbb{D}},$$ between Bloch-type and weighted Dirichlet-type spaces, where 𝜑 is a holomorphic self-map of 𝔻, n ∈ ℕ and g ∈ H(𝔻).

• Kyungpook Mathematical Journal
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• 제60권3호
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• pp.507-518
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• 2020

We consider a boundary version of the Schwarz Lemma on a certain class of functions which is denoted by 𝒩. For the function f(z) = z + a₂z² + a₃z³ + … which is defined in the unit disc D such that the function f(z) belongs to the class 𝒩, we estimate from below the modulus of the angular derivative of the function ${\frac{f{^{\prime}^{\prime}}(z)}{f(z)}}$ at the boundary point c with f'(c) = 0. The sharpness of these inequalities is also proved.

• East Asian mathematical journal
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• 제5권1호
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• pp.1-23
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• 1989

Let $S_p*({\alpha},{\beta},{\mu})$ denote the class of functions $f(z)=z^p-{\sum}{\limit}^{\infty}_{n=1}a_{p+n}\;z^{p+n}(a_{p+n}{\geq}o,\;p{\in}N)$ analytic and p-valent in the unit disc $U=\{z:{\mid}z{\mid}<1\}$ and satisfy the condition $${\mid}\frac{\frac{zf'(z)}{f(z)}-p}{\mu\frac{zf'(z)}{f(z)}+p-(1+\mu)\alpha}\mid<\beta,\;z{\in}U$$, where $o{\leq}{\alpha} and $o\leq\mu\leq1$. Further f(z) is said to belong to the class $C_p*({\alpha},{\beta},{\mu})\;if\;zf'(z)/p{\in}S_p*(\alpha,\beta,\mu)$. In this paper we obtain for these classes sharp results concerning coefficient estimates, disortion theorems, closure theorems, Hadamard products and some distortion theorems for the fractional calculus.