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LIMITS OF FRACTIONAL DERIVATIVES AND COMPOSITIONS OF ANALYTIC FUNCTIONS
- Part of
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- Journal:
- Journal of the Australian Mathematical Society / Volume 103 / Issue 1 / August 2017
- Published online by Cambridge University Press:
- 28 September 2016, pp. 104-115
- Print publication:
- August 2017
-
- Article
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- You have access
- Export citation
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Suppose that the function
$f$ is analytic in the open unit disk 
$\unicode[STIX]{x1D6E5}$ in the complex plane. For each 
$\unicode[STIX]{x1D6FC}>0$ a function 
$f^{[\unicode[STIX]{x1D6FC}]}$ is defined as the Hadamard product of 
$f$ with a certain power function. The function 
$f^{[\unicode[STIX]{x1D6FC}]}$ compares with the fractional derivative of 
$f$ of order 
$\unicode[STIX]{x1D6FC}$. Suppose that 
$f^{[\unicode[STIX]{x1D6FC}]}$ has a limit at some point 
$z_{0}$ on the boundary of 
$\unicode[STIX]{x1D6E5}$. Then 
$w_{0}=\lim _{z\rightarrow z_{0}}f(z)$ exists. Suppose that 
$\unicode[STIX]{x1D6F7}$ is analytic in 
$f(\unicode[STIX]{x1D6E5})$ and at 
$w_{0}$. We show that if 
$g=\unicode[STIX]{x1D6F7}(f)$ then 
$g^{[\unicode[STIX]{x1D6FC}]}$ has a limit at 
$z_{0}$.
