Let \begin{aligned} I=\int_0^1 B^{\frac{1}{1-\alpha + \alpha x}} x^{k - 1} \left(\frac{\alpha \log{\left(B \right)}}{(\alpha-k-1)^2} +\frac{1}{k} + \log{\left(x \right)} \right) dx, \end{aligned} where $B\geq 2, \alpha \in (0,1), k\geq 1$. Determine the sign of $I$.
Carlo Beenakker shows that for small 𝛼 it evaluates to
\begin{aligned} I= -\frac{\alpha^2 B\left(k^2+k-1\right)(2+\ln B) \ln B}{k(k+1)^3(k+2)^2}+\mathrm{O}\left(\alpha^3\right), \end{aligned}
I have tried numerical experiments and I find that they are not necessarily close to each other especially when $B$ is very large. Still, this is a good solution if we can prove that the asymptotic estimate is an upper bound for the real integration.
The problem in question actually originates from a control system. I have a function $T_k$ which can be expressed as:
\begin{equation} T_k = k \int_{0}^1 x^{k-1} B^{\frac{1}{1-\alpha + \alpha x}} \, dx. \end{equation}
My objective is to find the optimal value of $k$ that minimizes the entire system function.
However, evaluating the integral introduces considerable complexity, especially when analyzing the optimal point. As a result, I demonstrated that $ B^{\frac{1}{1-\alpha/(k+1)}}$ serves as a lower bound for this function. Subsequently, I established a series of theorems linking this lower bound to the actual integral.
My goal was to determine the ratio given by:
\begin{equation} u_k = k \int_{0}^1 x^{k-1} B^{\frac{1}{1-\alpha + \alpha x}-\frac{1}{1-\alpha/(k+1)}} \, dx. \end{equation}
This ratio exhibits monotonically decreasing behavior with respect to $k$. I differentiated this parameterized integral, transforming the problem into proving that its derivative is negative, implying that $I$ is less than 0.
I am sorry for bringing up this question again. When I posted it, I did not register, and I lost my right to even comment on the old question when I closed the question window.
