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.


Full Text





Article Information

TitleAnalyticity 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 $p$
SourceMethods Funct. Anal. Topology, Vol. 25 (2019), no. 4, 339-359
MilestonesReceived 17/11/2018; Revised 07/10/2019
CopyrightThe 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


Google Scholar Metrics

Citing articles in Google Scholar
Similar articles in Google Scholar

Export article

Save to Mendeley



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},
       URL = {http://mfat.imath.kiev.ua/article/?id=1242},
}


References

Coming Soon.

All Issues