Open Access

# Analyticity and other properties of functionals $I\left(f, p\right)=\int_{A}|f(t)|^p dt$ and $n(f,p)=\left(\frac{1}{\mu(A)}\int_{A}|f(t)|^p dt\right)^{\frac{1}{p}}$ as functions of variable $p$

### Abstract

We showed that for each function $f(t)$, which is not equal to zero almost everywhere in the Lebesgue measurable set, functionals $I\left(f,z\right)=\int_A{{|f(t)|}^z dt}$ as functions of a complex variable $z=p+iy$ are continuous on the domain and analytic on a set of all inner points of this domain. The functions $I(f,p)$ as functions of a real variable $p$ are strictly convex downward and log-convex on the domain. We proved that functionals $n(f,p)$ as functions of a real variable $p$ are analytic at all inner points of the interval, in which the function $n(f,p)\neq 0$ except the point $p=0$, continuous and strictly increasing on this interval.

Key words: Functional, approximation, analyticity, monotonicity, continuity.

### Article Information

 Title Analyticity and other properties of functionals $I\left(f, p\right)=\int_{A}|f(t)|^p dt$ and $n(f,p)=\left(\frac{1}{\mu(A)}\int_{A}|f(t)|^p dt\right)^{\frac{1}{p}}$ as functions of variable $p$ Source Methods Funct. Anal. Topology, Vol. 25 (2019), no. 4, 339-359 MathSciNet MR4049689 Milestones Received 17/11/2018; Revised 07/10/2019 Copyright The Author(s) 2019 (CC BY-SA)

### Authors Information

D. M. Bushev
Lesya Ukrainka Eastern European National University, 13 Volya Avenue, 43025, Lutsk, Ukraine

I. V. Kal'chuk
Lesya Ukrainka Eastern European National University, 13 Volya Avenue, 43025, Lutsk, Ukraine

### Citation Example

D. M. Bushev and I. V. Kal'chuk, Analyticity and other properties of functionals $I\left(f, p\right)=\int_{A}|f(t)|^p dt$ and $n(f,p)=\left(\frac{1}{\mu(A)}\int_{A}|f(t)|^p dt\right)^{\frac{1}{p}}$ as functions of variable $p$, Methods Funct. Anal. Topology 25 (2019), no. 4, 339-359.

### BibTex

@article {MFAT1242,
AUTHOR = {D. M. Bushev and I. V. Kal'chuk},
TITLE = {Analyticity and other properties of functionals  $I\left(f, p\right)=\int_{A}|f(t)|^p dt$ and $n(f,p)=\left(\frac{1}{\mu(A)}\int_{A}|f(t)|^p dt\right)^{\frac{1}{p}}$ as  functions of variable $p$},
JOURNAL = {Methods Funct. Anal. Topology},
FJOURNAL = {Methods of Functional Analysis and Topology},
VOLUME = {25},
YEAR = {2019},
NUMBER = {4},
PAGES = {339-359},
ISSN = {1029-3531},
MRNUMBER = {MR4049689},
URL = {http://mfat.imath.kiev.ua/article/?id=1242},
}

Coming Soon.