Communications of the Korean Mathematical Society (대한수학회논문집)

Korean Mathematical Society (대한수학회)
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ON COEFFICIENT PROBLEMS FOR STARLIKE FUNCTIONS RELATED TO VERTICAL STRIP DOMAINS

  • Received : 2018.03.16
  • Accepted : 2018.07.24
  • Published : 2019.04.30
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Abstract

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.

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DBSHCJ_2019_v34n2_451_f0001.png 이미지

FIGURE 1. Graphs of g, h, k on the interval [-1, x1]

DBSHCJ_2019_v34n2_451_f0002.png 이미지

FIGURE 2. Graphs of g, h, k on the interval [x1, x2]

DBSHCJ_2019_v34n2_451_f0003.png 이미지

FIGURE 3. Graphs of g, h, k on the interval [x2, x3]

DBSHCJ_2019_v34n2_451_f0004.png 이미지

FIGURE 4. Graphs of g, h, k on the interval [x3, x4]

DBSHCJ_2019_v34n2_451_f0005.png 이미지

FIGURE 5. Graphs of g, h, k on the interval [x4, 0]

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