Sharp bounds for initial coefficients and the second Hankel determinant
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For functions $f(z)=z+a_{2}z^{2}+a_{3}z^{3}+\cdots$ belonging to particular classes, this paper finds sharp bounds for the initial coefficients $a_{2}, \, a_{3}, \, a_{4},$ as well as the sharp estimate for the second order Hankel determinant $H_{2}(2)=a_{2}a_{4}-a_{3}^{2}.$ Two classes are treated: first is the class consisting of $f(z)=z+a_{2}z^{2}+a_{3}z^{3}+\cdots$ in the unit disk $\mathbb{D}$ satisfying $$\left|\left(\frac{z}{f(z)}\right)^{1+\alpha}f'(z)-1\right|<\lambda, \quad 0<\alpha <1, \, 0 < \lambda \leq 1.$$ The second class consists of Bazilevi\v{c} functions $f(z)=z+a_{2}z^{2}+a_{3}z^{3}+\cdots$ in $\mathbb{D}$ satisfying $${\rm Re}\left\{\left(\frac{f(z)}{z}\right)^{\alpha-1}f'(z)\right\}>0, \quad \alpha >0.$$
Abstract
For functions $f(z)=z+a_{2}z^{2}+a_{3}z^{3}+\cdots$ belonging to particular classes, this paper finds sharp bounds for the initial coefficients $a_{2}, \, a_{3}, \, a_{4},$ as well as the sharp estimate for the second order Hankel determinant $H_{2}(2)=a_{2}a_{4}-a_{3}^{2}.$ Two classes are treated: first is the class consisting of $f(z)=z+a_{2}z^{2}+a_{3}z^{3}+\cdots$ in the unit disk $\mathbb{D}$ satisfying $$\left|\left(\frac{z}{f(z)}\right)^{1+\alpha}f'(z)-1\right|<\lambda, \quad 0<\alpha <1, \, 0 < \lambda \leq 1.$$ The second class consists of Bazilevi\v{c} functions $f(z)=z+a_{2}z^{2}+a_{3}z^{3}+\cdots$ in $\mathbb{D}$ satisfying $${\rm Re}\left\{\left(\frac{f(z)}{z}\right)^{\alpha-1}f'(z)\right\}>0, \quad \alpha >0.$$
참고문헌 (14)
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