Polyak in his book Introduction to Optimization (the book is freely downloadable at his personal web page here, year 1987) states at page 25 the following theorem
Let $f(x)$ be twice differentiable and let $$ I \ell \le \nabla^2 f(x) \le IL, \quad \ell > 0, \ L \in \mathbb{R} $$ for all $x$. Then for all $0 < \alpha < 2/ L$ $$ || x_k - x^* || \le ||x_0 - x^* || q^k , \quad q:= \max \{ |1- \alpha \ell |,|1- \alpha L | \} < 1$$
where $x^*$ is the unique minimum of the function $f(x)$ and $x_k$ is the $k$-th step of the gradient descent scheme
$$ x_{k+1} = x_k + \alpha \nabla f(x_k) $$ starting at $x_0$.
I am looking for a result of this kind ( where the convergence of gradient descent is proven for $x_k \rightarrow x^*$ and not $f(x_k) \rightarrow f(x^*)$ ) in the case that the objective function $f(x)$ is not strongly convex but just strictly convex or convex, i.e., when
$$ 0 \le \nabla^2 f(x) \le IL, \quad \ L \in \mathbb{R} $$