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Results & Lemmas (5)
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Theorem 2.
Theorem 2..
Theorem 2. $B(\alpha, \beta)\subseteqq K(\alpha, \beta)$.
Theorem 3 · coeff
Theorem 3 [Coefficient estimates]. If is in, then with equality only for functions of the form (2.4).
Theorem 3 [Coefficient estimates]. If $f(z)=z-\sum_{*=2}^{\infty}|a_{n}|z$ is in $B(\alpha, \beta)$, then $|a_{n}|\leqq\frac{2\beta(1-\alpha\rangle n+(1-\beta+2\alpha\beta)}{(1+\beta)n^{2}}$ with equality only for functions of the form (2.4) $f_{n}(z)=z-\frac{2\beta(1-\alpha)n+(1-\beta+2\alpha\beta)}{(1+\beta)n^{2}}z^{n}$ .
Theorem 4
Theorem 4 [Distortion Theorem]. If, then for, (2.5). (2.6).
Theorem 4 [Distortion Theorem]. If $f\in B(\alpha, \beta)$, then for $|z|\leqq r<1$, (2.5) $r-\frac{1+3\beta-2a\beta}{4(1+\beta)}r\leqq|f(z)|\leqq r+\frac{1+3\beta-2\alpha\beta}{4(1+\beta)}r^{2}$ . (2.6) $1-\frac{1+3\beta-2\alpha\beta}{2(1+\beta)}r\leqq|f^{\prime}(z)|\leqq 1+\frac{1+3\beta-2a\beta}{2(1+\beta)}r$ .
Theorem 5.
Theorem 5. Let. Then the disk is maPped onto a domain that contains the disk. The result is sharp with extremal function defined in (2.7).
Theorem 5. Let $f\in B(a, \beta)$. Then the disk $|z|<1$ is maPped onto a domain that contains the disk $|w|<(3+\beta+2\alpha\beta/4(1+\beta))$. The result is sharp with extremal function defined in (2.7).
Theorem 6.
Theorem 6. The class is convex.
Theorem 6. The class $B(\alpha, \beta)$ is convex.
Function classes studied:
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