• Journal of applied mathematics & informatics
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• v.2 no.2
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• pp.83-95
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• 1995

Given a closed subset of the family $S^{*}(\alpha)$ of functions starlike of order $\alpha$, a continuous Frechet differentiable functional J, is constructed with this collection as the solution set to the extremal problem ReJ(f) over $S^{*}(\alpha)$. The support points of $S^{*}(\alpha)$ is completely characterized and shown to coincide with the extreme points of its convex hulls. Given any finite collection of support points of $S^{*}(\alpha)$ a continuous linear functional J, is constructed with this collection as the solution set to the extremal problem ReJ(f) over $S^{*}(\alpha)$.

• Kyungpook Mathematical Journal
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• v.59 no.2
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• pp.215-224
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• 2019

In the present paper, some generalizations of analytic functions have been considered and the bounds of the coefficients of these classes of bi-univalent functions have been investigated.

• Journal of the Korean Mathematical Society
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• v.43 no.2
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• pp.311-322
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• 2006

For the Briot-Bouquet differential equations of the form given in [1] $${{\mu}(z)+\frac {z{\mu}'(z)}{z\frac {f'(z)}{f(z)}\[\alpha{\mu}(z)+\beta]}=g(z)$$ we can reduce them to $${{\mu}(z)+F(z)\frac {v'(z)}{v(z)}=h(z)$$ where $$v(z)=\alpha{\mu}(z)+\beta,\;h(z)={\alpha}g(z)+\beta\;and\;F(z)=f(z)/f'(z)$$. In this paper we are going to give conditions in order that if u and v satisfy, respectively, the equations (1) $${{\mu}(z)+F(z)\frac {v'(z)}{v(z)}=h(z)$$, $${{\mu}(z)+G(z)\frac {v'(z)}{v(z)}=g(z)$$ with certain conditions on the functions F and G applying the concept of strong subordination $g\;\prec\;\prec\;h$ given in [2] by the author, implies that $v\;\prec\;{\mu},\;where\;\prec$ indicates subordination.

• Journal of the Korean Mathematical Society
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• v.55 no.5
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• pp.1019-1030
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• 2018

In [12,13] the researchers introduced universally convex, universally starlike and universally prestarlike functions in the slit domain ${\mathbb{C}}{\backslash}[1,{\infty})$. These papers extended the corresponding notions from the unit disc to other discs and half-planes containing the origin. In this paper, we introduce universally prestarlike generalized functions of order ${\alpha}$ with ${\alpha}{\leq}1$ and we obtain upper bounds of the second Hankel determinant ${\mid}a_2a_4-a^2_3{\mid}$ for such functions.

• Kyungpook Mathematical Journal
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• v.45 no.3
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• pp.395-404
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• 2005

Let α be a complex number with 𝕽α > 0. Let the functions f and g be analytic in the unit disc E = {z : |z| < 1} and normalized by the conditions f(0) = g(0) = 0, f'(0) = g'(0) = 1. In the present article, we study the differential subordinations of the forms $${\alpha}{\frac{z^2f^{{\prime}{\prime}}(z)}{f(z)}}+{\frac{zf^{\prime}(z)}{f(z)}}{\prec}{\alpha}{\frac{z^2g^{{\prime}{\prime}}(z)}{g(z)}}+{\frac{zg^{\prime}(z)}{g(z)}},\;z{\in}E,$$ and $${\frac{z^2f^{{\prime}{\prime}}(z)}{f(z)}}{\prec}{\frac{z^2g^{{\prime}{\prime}}(z)}{g(z)}},\;z{\in}E.$$ As consequences, we obtain a number of sufficient conditions for star likeness of analytic maps in the unit disc. Here, the symbol ' ${\prec}$ ' stands for subordination

• The Journal of the Petrological Society of Korea
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• v.28 no.1
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• pp.1-13
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• 2019

The characteristics of the rock cleavage of Jurassic Geochang granite were analysed using the parameters from the length and spacing-cumulative frequency diagrams. The evaluation for three planes and three rock cleavages was performed using the 25 parameters such as (1~2) slope angle(${\alpha}^{\circ}$and ${\beta}^{\circ}$), (3) intersection angle(${\alpha}-{\beta}^{\circ}$), (4) exponent difference(${\lambda}_S-{\lambda}_L$), (5~12) length of line(oa, ob, ol, os, ss', ll' and sl') and (13~15) length ratio(ol/os, ss'/ll' and ll'/sl'), (16) mean length((ss'+ll')/2), (17~23) area (${\Delta}oaa^{\prime}$, ${\Delta}obb^{\prime}$, ${\Delta}obb^{\prime}$, ${\Delta}oaa_a^{\prime}$, ${\Delta}obb_a^{\prime}$, ${\Delta}ll^{\prime}s^{\prime}$, ${\Delta}ss^{\prime}l^{\prime}$ and ⏢$ll^{\prime}ss^{\prime}$) and (24~25) area difference(${\Delta}obb^{\prime}-{\Delta}oaa^{\prime}$ and ${\Delta}obb_a^{\prime}-{\Delta}oaa_a^{\prime}$). Firstly, the values of the 11 parameters(group I: No. 1, 3~4, 7, 9~10, 13, 15~16, 20 and 25), the 3 parameters(group II: No. 5, 8 and 17) and the 2 parameters(group III: No. 12 and 22) are in orders of H(hardway) < G(grain) < R(rift), R < G < H and G < H < R, respectively. On the contrary, the values of parameters belonging to the above three groups show reverse orders for three planes. Secondly, the generalized chart for three planes and three rock cleavages were made. From the related chart, the distribution types formed by the two diagrams related to lengths and spacings were derived. The diagrams related to spacings show upward curvature in the chart of rift plane(G1 & H1, R') and hardway(H1 & H2, H). On the contrary, the diagrams related to lengths show downward curvature. These two diagrams take the form of a convex lens in the upper section. Besides, the two diagrams cross each other in the lower section. The overall shape formed by the above two diagrams between three planes($H^{\prime}{\rightarrow}G^{\prime}{\rightarrow}R^{\prime}$) and three rock cleavages($R{\rightarrow}G{\rightarrow}H$) display in reverse order. Lastly, these types of correlation analysis is useful for discriminating three quarrying planes.