DOI: 10.68381/jca20015 ISSN: 0944-6532
Convex Conjugates of Analytic Functions of Logarithmically Convex Functionals
Krzysztof Zajkowski
Let
f_{\bf c}(r)=\sum_{n=0}^\infty e^{c_n}r^n
f
c
(
r
)
=
∑
n
=
0
∞
e
c
n
r
n
be an analytic function;
{\bf c}=(c_n)\in l_\infty
c
=
(
c
n
)
∈
l
∞
. We assume that
r
r
is some logarithmically convex and lower semicontinuous functional on a locally convex topological space
L
L
. In this paper we derive a formula on the Legendre-Fenchel transform of a functional
\widehat{\lambda}({\bf c},\varphi)= \ln f_{\bf c}(e^{\lambda(\varphi)})\,
λ
^
(
c
,
φ
)
=
ln
f
c
(
e
λ
(
φ
)
)
where
\lambda(\varphi)=\ln r(\varphi)
λ
(
φ
)
=
ln
r
(
φ
)
(
\varphi\in L
φ
∈
L
). In this manner we generalize to the infinite case Theorem 3.1 of the paper of U. Ostaszewska and K. Zajkowski ["Legendre-Fenchel transform of the spectral exponent of polynomials of weighted composition operators", Positivity, DOI 10.1007/s11117-009-0023-6].