New posts in real-analysis

Suppose $1\le p < r < q < \infty$. Prove that $L^p\cap L^q \subset L^r$.

Let $f$ be a cont. on $\mathbb{R}$ and define $G(x)=\int_0^{\sin (x)}f(t) dt $. Show that $G$ is differentiable on $\Bbb{R}$ and compute $G'$.

Is this sequence bounded ? (An open problem between my schoolmates !)

Show $\sigma_{X}^{2}(t)=\begin{cases} x_{0}\frac{\beta}{\alpha}e^{\alpha t}[e^{\alpha t}-1], & \alpha \neq 0\\ x_{0}\beta t, & \alpha = 0 \end{cases}$

To find continuous functions on $\mathbb R$ which preserve certain algebraic structures

$U\subset [0,\infty)$ is open and unbounded $\Rightarrow \exists x$ such that $U\cap \{nx;n\in \mathbb N\}$ is infinite.

How to show that the series of $\frac{\sin(n)}{\log(n)}$ converges?

A smooth nowhere analytic function such that all derivatives are monotone

Proving algebraically $a^2+b^2\ge a^{\alpha}b^{2-\alpha}$ for $0\le\alpha\le2$ and $a,b\ge0$

Function satisfying $\lim_{x\to 0}\frac{f(ax)}{f(x)}=a$

Is it possible that $\sum a_{\sigma(n)}$ converges iff $\sum b_{\sigma(n)}$ diverges for every permutation $\sigma :\mathbb N\to \mathbb N$?

Comparison of $L^2$ and $L^1$ norms for functions

understand what ∪n∈NUn = (0, 2) ⊃ (0, 1] means [closed]

Second derivative: how should one think about?

Why do we need min to choose $\delta$?

convergence of $\sum_{n=1}^\infty \frac{n^{-x} + 1}{|log(x)|^n +1}$ if $x>0$

Find $\int_0^1 \frac{x-1}{\ln (x)} \, dx$

Are there functions $f(t)$ with $||f'(t)||_\infty < \infty$ such as their Fourier transform $F(w)$ makes $\int_{-\infty}^\infty|wF(w)|dw \to \infty$??

Towards a Little proof of Fermat's last theorem

Convex functions in integral inequality