Hardy's and related inequalities in quotients (Q2921637)

The authors generalize some classic inequalities in the following way.NEWLINENEWLINEHardy's: Let \(I\subset \mathbb {R}\) be an interval and \(\Phi \:I\to \mathbb {R}\) be a positive convex function. Assume further that \(0<p\leq q<\infty \) and \(u\:(0,\infty)\to \mathbb {R}\) is a weight function. Then for all measurable functions \(f_1,f_2 : (0,\infty)\to \mathbb {R}\) such that \(\frac {f_1}{f_2}(y)\in I\) and NEWLINE\[NEWLINE v(y)=f_2(y)\left (\int \limits_y^\infty \left (\int \limits_0^x f_2(\tau){\operatorname {d}}\tau \right)^{\!\!\!\!-\frac {q}{p}}u(x)\,{\operatorname {d}}x\right)^{\!\!\!\frac {p}{q}} NEWLINE\]NEWLINE the inequality NEWLINE\[NEWLINE \left (\int \limits_0^\infty u(x)\left [\Phi \left (\frac {\int \limits_0^x f_1(y) {\operatorname {d}}y}{\int \limits_0^x f_2(y){\operatorname {d}}y} \right)\right]^{\frac {q}{p}}\!{\operatorname {d}}x\right)^{\!\!\!\frac {1}{q}}\leq \left (\int \limits_0^\infty v(y)\,\Phi \left (\frac {f_1(y)}{f_2(y)}\right){\operatorname {d}}y\right)^{\!\!\!\frac {1}{p}} NEWLINE\]NEWLINE holds. Hardy-Hilbert's:NEWLINENEWLINEWith the same assumptions, \(s\in \mathbb {R}\) and NEWLINE\[NEWLINE v(y)= f_2(y)\left (\int \limits_0^\infty \frac {u(x)}{(x+y)^{\frac {sq}{p}}}\left (\int \limits_0^x \frac {f_2(\tau)}{(x+\tau)^{s}}{\operatorname {d}}\tau \right)^{\!\!\!\!-\frac {q}{p}}{\operatorname {d}}x\right)^{\!\!\!\frac {p}{q}} NEWLINE\]NEWLINE one has NEWLINE\[NEWLINE \left (\displaystyle \int \limits_0^\infty u(x)\left [\Phi \left (\frac {\displaystyle \int \limits_0^x \frac {f_1(y)}{(x+y)^{s}}\!{\operatorname {d}}y}{\displaystyle \int \limits_0^x \frac {f_2(y)}{(x+y)^{s}}{\operatorname {d}}y} \right)\right]^{\frac {q}{p}}\!{\operatorname {d}}x\right)^{\!\!\!\frac {1}{q}}\leq \left (\displaystyle \int \limits_0^\infty v(y)\,\Phi \left (\frac {f_1(y)}{f_2(y)}\right){\operatorname {d}}y\right)^{\!\!\!\frac {1}{p}}. NEWLINE\]NEWLINENEWLINENEWLINEHardy-Littlewood-Pólya's: With NEWLINE\[NEWLINE v(y)= f_2(y)\left (\int \limits_0^\infty \frac {u(x)}{\max (x,y)^{\frac {sq}{p}}} \left (\int \limits_0^x \frac {f_2(\tau)}{\max (x,\tau)^{s}}{\operatorname {d}}\tau \right)^{\!\!\!\!-\frac {q}{p}}\!{\operatorname {d}}x\right)^{\!\!\!\frac {p}{q}} NEWLINE\]NEWLINE one has NEWLINE\[NEWLINE \left (\displaystyle \int \limits_0^\infty u(x)\left [\Phi \left (\frac {\displaystyle \int \limits_0^x \frac {f_1(y)}{\max (x,y)^{s}}{\operatorname {d}}y}{\displaystyle \int \limits_0^x \frac {f_2(y)}{\max (x,y)^{s}}{\operatorname {d}}y} \right)\right]^{\frac {q}{p}}\!{\operatorname {d}}x\right)^{\!\!\frac {1}{q}}\!\leq \left (\displaystyle \int \limits_0^\infty v(y)\,\Phi \left (\frac {f_1(y)}{f_2(y)}\right){\operatorname {d}}y\right)^{\!\!\frac {1}{p}}. NEWLINE\]NEWLINENEWLINENEWLINEHardy-Hilbert type: With \(L(u,v)=\frac {u-v}{\log u-\log v}\) and NEWLINE\[NEWLINE v(y)= f_2(y)\left (\int \limits_0^\infty \frac {u(x)}{L(x,y)^{\frac {q}{p}}}\left (\int \limits_0^x \frac {f_2(\tau)}{L(x,\tau)}{\operatorname {d}}\tau \right)^{\!\!\!\!-\frac {q}{p}}\!{\operatorname {d}}x\right)^{\!\!\!\frac {p}{q}} NEWLINE\]NEWLINE one has NEWLINE\[NEWLINE \left (\displaystyle \int \limits_0^\infty u(x)\left [\Phi \left (\frac {\displaystyle \int \limits_0^x \frac {f_1(y)}{L(x,y)}{\operatorname {d}}y}{\displaystyle \int \limits_0^x \frac {f_2(y)}{L(x,y)}{\operatorname {d}}y} \right)\right]^{\frac {q}{p}}\!{\operatorname {d}}x\right)^{\!\!\!\frac {1}{q}}\!\leq \left (\displaystyle \int \limits_0^\infty v(y)\Phi \left (\frac {f_1(y)}{f_2(y)}\right){\operatorname {d}}y\right)^{\!\!\frac {1}{p}}. NEWLINE\]NEWLINE Numerous examples are provided.