6,114 research outputs found
A comprehensive class of harmonic functions defined by convolution and its connection with integral transforms and hypergeometric functions
For given two harmonic functions and with real coefficients in
the open unit disk , we study a class of harmonic functions
satisfying \RE \frac{(f*\Phi)(z)}{(f*\Psi)(z)}>\alpha \quad (0\leq
\alpha <1, z \in \mathbb{D}); * being the harmonic convolution. Coefficient
inequalities, growth and covering theorems, as well as closure theorems are
determined. The results obtained extend several known results as special cases.
In addition, we study the class of harmonic functions that satisfy \RE
f(z)/z>\alpha . As an application, their
connection with certain integral transforms and hypergeometric functions is
established.Comment: 14pages, 1 figur
The extended hypergeometric class of L\'evy processes
With a view to computing fluctuation identities related to stable processes,
we review and extend the class of hypergeometric L\'evy processes explored in
Kuznetsov and Pardo (arXiv:1012.0817). We give the Wiener-Hopf factorisation of
a process in the extended class, and characterise its exponential functional.
Finally, we give three concrete examples arising from transformations of stable
processes.Comment: 22 page
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