Titles and Short Summaries of the Talks

33 Complex Analysis
9:30–12:00
Complex Analysis
March 22nd (Fri) Conference Room VIII
Final: 2013/2/7
1 Katsuyuki Nishimoto
∗ 2 −3 N-fractional calculus of the function f(z) = ((z−b) −c) and identities
(Descartes Press Co.) · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · 15
Summary: In this article the N-fractional calculus of the function in title is discussed by the
calculation in the manners (((z − b) 2 − c) −4 · (z − b) 2 − c)r and ((((z − b) 2 − c) −3 )1)r−1. Moreover
some identities derived from them are presented.
One of them is shown as follows for example, ∑ ∞
2γG(k, γ, 1)+γ(γ −1)G(k, γ, 2) where G(k, γ, m) = ∑ ∞
k=0
[3]kΓ(2k+6+γ)
k=0
[4]kΓ(2k+8+γ−m)
k!Γ(2k+8)
k!Γ(2k+6) Sk = (1 − S)G(k, γ, 0) −
Sk , (|Γ(2k +8+γ −m)| <
∞) S = c
(z−b) 2 , |S| < 1, Γ(· · · ); Gamma function and [λ]k = λ(λ + 1) · · · (λ + k − 1) = Γ(λ + k)/Γ(λ)
with [λ0] = 1. (Notation of Pochhammer)
2 Mitsuru Uchiyama (Shimane Univ.) ♯ Principal inverses of orthogonal polynomials · · · · · · · · · · · · · · · · · · · · · · · 15
Summary: Let {pn(t)} be orthogonal polynomials with positive leading coefficients and dn the
maximal zero of p ′ n(t). The inverse of the restriction of pn(t) to [dn, ∞) is called the principal
inverse of pn and denoted by p −1
n . We show it has an analytic continuation to the upper half plane
that is a Pick (or Nevanlinna) function.
3 Hitoshi Shiraishi (Kinki Univ.) ♯ Coefficient estimates for Schwarz functions · · · · · · · · · · · · · · · · · · · · · · · · 15
Toshio Hayami (Kinki Univ.)
Summary: Let B be the class of functions w(z) of the form w(z) = ∞∑
bkzk which are analytic and
|w(z)| < 1 in the open unit disk U = {z ∈ C : |z| < 1}. Then we call w(z) ∈ B the Schwarz function.
Also, let P dnote the class of functions p(z) of the form p(z) = 1 + ∞∑
ckzk which are analytic and
Re (p(z)) > 0 in U. In the present talk, we discuss new coefficient estimates for Schwarz functions.
4 Toshio Hayami (Kinki Univ.) ♯ Coefficient estimates for a certain class concerned with arguments of
Shigeyoshi Owa (Kinki Univ.) f ′ (z) · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · 15
Summary: Let A denote the class of functions f(z) of the form f(z) = z + ∑ ∞
n=2 anz n which are
analytic in the open unit disk U = {z ∈ C : |z| < 1}, and let P be the class of functions p(z) of the
form p(z) = 1 + ∑ ∞
k=1 ckz k which are analytic in U and satisfy the condition Re (p(z)) > 0 (z ∈ U).
In the present talk, we discuss the coefficient estimates of functions f(z) ∈ A satisfying f ′ (z0) = 0
for some z0 ∈ U.
5 Junichi Nishiwaki (Setsunan Univ.) ♯ Notes on a certain class of analytic functions · · · · · · · · · · · · · · · · · · · · · · 15
Shigeyoshi Owa (Kinki Univ.)
Summary: Let A be the class of analytic functions f(z) in the open unit disk U. Furthermore, the
subclass B of A concerned with the class of uniformly convex functions or the class Sp is defined.
By virtue of some properties of uniformly convex functions and the class Sp, some properties of the
class B are considered.
k=1
k=1