Atlas: Integral means inequalities for fractional calculus operator by Tadayuki Sekine

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3rd International ISAAC Congress
August 20-25, 2001
Freie Universitaet Berlin
Berlin, Germany

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Integral means inequalities for fractional calculus operator
by
Tadayuki Sekine
Nihon University, Japan

Let A denote the class of functions f( z) normalized by

f( z) = z+

\infty
å
k=2

a_kz^k,

which are analytic in the open unit disk U.

Denote by A( n) the subclass of A consisting of all functions f( z) of the form:

f( z) = z- \infty
å
k=n+1
a_kz^k (a_k\geqq 0;n in N).

We denote by T( n) the subclass of A( n) of functions which are univalent in U, and by T_\alpha( n) and C_\alpha( n) the subclasses of T(n) consisting of functions which are, respectively, starlike of order \alpha ( 0\leqq \alpha < 1) and convex of order \alpha ( 0\leqq \alpha < 1) .

We denote by A(n, \vartheta) the subclass of A consisting of all functions f( z) of the form :

f( z) = z- \infty
å
k=n+1
e^{i( k-1) \vartheta}a_kz^k (\vartheta in R;a_k\geqq 0;n in N).
(\theequation)
We also define the subclasses T(n, \vartheta), T_\alpha^*(n, \vartheta) and C_\alpha(n, \vartheta) of the class A(n, \vartheta) in the same way as we defined the subclasses T(n), T_\alpha(n) and C_\alpha(n) of the class A( n) .

We introduce a general family A( n;{ B_k}, \vartheta) of functions f in A( n, \vartheta) of the form (1), which satisfy the following inequality:

\infty
å
k=n+1
B_ka_k\leqq 1 ( B_k > 0, n in N)

for some positive sequence { B_k} of real numbers.

We show several integral means inequalities for fractional calculus operator of functions belonging to above generalized family.

Date received: June 5, 2001