대한수학회지 (Journal of the Korean Mathematical Society)

  • 제52권4호
  • /
  • Pages.727-749
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  • 2015
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  • 0304-9914(pISSN)
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  • 2234-3008(eISSN)
대한수학회 (Korean Mathematical Society)
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SUFFICIENT CONDITIONS FOR STARLIKENESS

  • 투고 : 2014.07.31
  • 발행 : 2015.06.01
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초록

We obtain the conditions on ${\beta}$ so that $1+{\beta}zp^{\prime}(z){\prec}1+4z/3+2z^2/3$ implies p(z) ${\prec}$ (2+z)/(2-z), $1+(1-{\alpha})z$, $(1+(1-2{\alpha})z)/(1-z)$, ($0{\leq}{\alpha}$<1), exp(z) or ${\sqrt{1+z}}$. Similar results are obtained by considering the expressions $1+{\beta}zp^{\prime}(z)/p(z)$, $1+{\beta}zp^{\prime}(z)/p^2(z)$ and $p(z)+{\beta}zp^{\prime}(z)/p(z)$. These results are applied to obtain sufficient conditions for normalized analytic function f to belong to various subclasses of starlike functions, or to satisfy the condition ${\mid}log(zf^{\prime}(z)/f(z)){\mid}$ < 1 or ${\mid}(zf^{\prime}(z)/f(z))^2-1{\mid}$ < 1 or zf'(z)/f(z) lying in the region bounded by the cardioid $(9x^2+9y^2-18x+5)^2-16(9x^2+9y^2-6x+1)=0$.

키워드

참고문헌

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피인용 문헌

  1. Subordinations for Functions with Positive Real Part 2017, https://doi.org/10.1007/s11785-017-0690-4