<abstract xmlns="http://www.w3.org/1999/xhtml">Given a continuous strictly monotonic real-valued function α, defined on an interval <italic>I</italic>, and a function <italic>ω</italic> : <italic>I →</italic> (0, +∞) we denote by <italic>B</italic>α<italic>ω</italic> the Bajraktarević mean generated by α and weighted by <italic>ω</italic>:<disp-formula><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_amsil-2020-0012_eq_001.png"></graphic><math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mrow><msubsup><mi>B</mi><mi>ω</mi><mi>α</mi></msubsup><mrow><mo>(</mo><mrow><mi>x</mi><mo>,</mo><mi>y</mi></mrow><mo>)</mo></mrow><mo>=</mo><msup><mi>α</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup><mrow><mo>(</mo><mrow><mfrac><mrow><mi>ω</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow></mrow><mrow><mi>ω</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>+</mo><mi>ω</mi><mrow><mo>(</mo><mi>y</mi><mo>)</mo></mrow></mrow></mfrac><mi>α</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>+</mo><mfrac><mrow><mi>ω</mi><mrow><mo>(</mo><mi>y</mi><mo>)</mo></mrow></mrow><mrow><mi>ω</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>+</mo><mi>ω</mi><mrow><mo>(</mo><mi>y</mi><mo>)</mo></mrow></mrow></mfrac><mi>α</mi><mrow><mo>(</mo><mi>y</mi><mo>)</mo></mrow></mrow><mo>)</mo></mrow><mo>,</mo><mi> </mi><mi> </mi><mi> </mi><mi>x</mi><mo>,</mo><mi>y</mi><mo>∈</mo><mi>I</mi><mo>.</mo></mrow></math><tex-math>B_\omega ^\alpha \left({x,y} \right) = {\alpha ^{- 1}}\left({{{\omega \left(x \right)} \over {\omega \left(x \right) + \omega \left(y \right)}}\alpha \left(x \right) + {{\omega \left(y \right)} \over {\omega \left(x \right) + \omega \left(y \right)}}\alpha \left(y \right)} \right),\,\,\,x,y \in I.</tex-math></alternatives></disp-formula>We find a necessary integral formula for all possible three times differentiable solutions (<italic>φ, ψ</italic>) of the functional equation<disp-formula><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_amsil-2020-0012_eq_002.png"></graphic><math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mrow><mi>r</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><msubsup><mi>B</mi><mi>s</mi><mi>ϕ</mi></msubsup><mrow><mo>(</mo><mrow><mi>x</mi><mo>,</mo><mi>y</mi></mrow><mo>)</mo></mrow><mo>+</mo><mi>r</mi><mrow><mo>(</mo><mi>y</mi><mo>)</mo></mrow><msubsup><mi>B</mi><mi>t</mi><mi>ψ</mi></msubsup><mrow><mo>(</mo><mrow><mi>x</mi><mo>,</mo><mi>y</mi></mrow><mo>)</mo></mrow><mo>=</mo><mi>r</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mi>x</mi><mo>+</mo><mi>r</mi><mrow><mo>(</mo><mi>y</mi><mo>)</mo></mrow><mi>y</mi><mo>,</mo></mrow></math><tex-math>r\left(x \right)B_s^\varphi \left({x,y} \right) + r\left(y \right)B_t^\psi \left({x,y} \right) = r\left(x \right)x + r\left(y \right)y,</tex-math></alternatives></disp-formula>where <italic>r, s, t</italic> : <italic>I →</italic> (0, +∞) are three times differentiable functions and the first derivatives of <italic>φ, ψ</italic> and <italic>r</italic> do not vanish. However, we show that not every pair (<italic>φ, ψ</italic>) given by the found formula actually satisfies the above equation.</abstract>

Given a continuous strictly monotonic real-valued function α, defined on an interval I, and a function ω : I → (0, +∞) we denote by Bαω the Bajraktarević mean generated by α and weighted by ω:Bωα(x,y)=α-1(ω(x)ω(x)+ω(y)α(x)+ω(y)ω(x)+ω(y)α(y)),   x,y∈I.B_\omega ^\alpha \left({x,y} \right) = {\alpha ^{- 1}}\left({{{\omega \left(x \right)} \over {\omega \left(x \right) + \omega \left(y \right)}}\alpha \left(x \right) + {{\omega \left(y \right)} \over {\omega \left(x \right) + \omega \left(y \right)}}\alpha \left(y \right)} \right),\,\,\,x,y \in I.We find a necessary integral formula for all possible three times differentiable solutions (φ, ψ) of the functional equationr(x)Bsϕ(x,y)+r(y)Btψ(x,y)=r(x)x+r(y)y,r\left(x \right)B_s^\varphi \left({x,y} \right) + r\left(y \right)B_t^\psi \left({x,y} \right) = r\left(x \right)x + r\left(y \right)y,where r, s, t : I → (0, +∞) are three times differentiable functions and the first derivatives of φ, ψ and r do not vanish. However, we show that not every pair (φ, ψ) given by the found formula actually satisfies the above equation.

On a Functional Equation Appearing on the Margins of a Mean Invariance Problem

Annales Mathematicae Silesianae

This work is licensed under the Creative Commons Attribution 4.0 International License.

{"article-title":"On a Functional Equation Appearing on the Margins of a Mean Invariance Problem"}

Given a continuous strictly monotonic real-valued function α, defined on an interval I, and a function ω : I → (0, +∞) we denote by Bαω the Bajraktarević mean...

On a Functional Equation Appearing on the Margins of a Mean Invariance Problem

Data publikacji: 09 lip 2020

Zakres stron: 96 - 103

Otrzymano: 26 gru 2019

Przyjęty: 03 cze 2020

DOI: https://doi.org/10.2478/amsil-2020-0012

Słowa kluczowe
functional equation, mean, invariance, Bajraktarević mean

© 2020 Justyna Jarczyk et al., published by Sciendo

This work is licensed under the Creative Commons Attribution 4.0 International License.

On a Functional Equation Appearing on the Margins of a Mean Invariance Problem

Data publikacji: 09 lip 2020

Zakres stron: 96 - 103

Otrzymano: 26 gru 2019

Przyjęty: 03 cze 2020

DOI: https://doi.org/10.2478/amsil-2020-0012

Słowa kluczowefunctional equation, mean, invariance, Bajraktarević mean

© 2020 Justyna Jarczyk et al., published by Sciendo

This work is licensed under the Creative Commons Attribution 4.0 International License.

Słowa kluczowe
functional equation, mean, invariance, Bajraktarević mean