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- Publisher
- Springer Journals
- Copyright
- Copyright © 2016 by Malaysian Mathematical Sciences Society and Penerbit Universiti Sains Malaysia
- Subject
- Mathematics; Mathematics, general; Applications of Mathematics
- ISSN
- 0126-6705
- eISSN
- 2180-4206
- DOI
- 10.1007/s40840-016-0414-3
- Publisher site
- See Article on Publisher Site
Abstract
Let $$\varphi $$ φ be an analytic function with the positive real parts, $$\varphi (0)= 1$$ φ ( 0 ) = 1 and $$\varphi ^{\prime }(0) >0$$ φ ′ ( 0 ) > 0 . Let $$f(z)= z+ a_2 z^2 +a_3 z^3 +{\cdots }$$ f ( z ) = z + a 2 z 2 + a 3 z 3 + ⋯ be an analytic function satisfying the subordination $$\alpha f^{\prime }(z) + (1-\alpha )zf^{\prime }(z)/f(z) \prec \varphi (z)$$ α f ′ ( z ) + ( 1 - α ) z f ′ ( z ) / f ( z ) ≺ φ ( z ) , $$(f^{\prime }(z))^{\alpha } (zf^{\prime }(z)/f(z))^{(1-\alpha )} \prec \varphi (z)$$ ( f ′ ( z ) ) α ( z f ′ ( z ) / f ( z ) ) ( 1 - α ) ≺ φ ( z ) , $$(f^{\prime }(z))^{\alpha } (1+ zf^{\prime \prime }(z)/f^{\prime }(z))^{(1-\alpha )} \prec \varphi (z)$$ ( f ′ ( z ) ) α ( 1 + z f ″ ( z ) / f ′ ( z ) ) ( 1 - α ) ≺ φ ( z ) , $$ (f(z)/z)^{\alpha } (zf^{\prime }(z)/f(z))^{(1-\alpha )} \prec \varphi (z)$$ ( f ( z ) / z ) α ( z f ′ ( z ) / f ( z ) ) ( 1 - α ) ≺ φ ( z ) , or $$(f(z)/z)^{\alpha } (1+ zf^{\prime \prime }(z)/f^{\prime }(z))^{(1-\alpha )} \prec \varphi (z)$$ ( f ( z ) / z ) α ( 1 + z f ″ ( z ) / f ′ ( z ) ) ( 1 - α ) ≺ φ ( z ) . For these functions, the bounds for the second Hankel determinant $$a_2a_4-a_3^2$$ a 2 a 4 - a 3 2 as well as the Fekete–Szegö coefficient functional are obtained. Our results include some previously known results.
Journal
Bulletin of the Malaysian Mathematical Sciences Society – Springer Journals
Published: Sep 30, 2016