Let $k\geq 1$. Consider Taylor's theorem. We know the Peano form and the mean-value form of the remainder term:
Peano form of the remainder
Let $f\colon (-\varepsilon,\varepsilon)\to\mathbb R$ be $k$ times differentiable. Then it holds \[ f(h) = \sum_{m=0}^k \frac{f^{(m)}(0)h^m}{m!} + o(h^k). \]
Mean-value form of the remainder
Let $f\colon (-\varepsilon,\varepsilon)\to\mathbb R$ be $k+1$ times differentiable. Then it holds \[ f(h) = \sum_{m=0}^k \frac{f^{(m)}(0)h^m}{m!} + \mathcal O(h^{k+1}). \]
Another form of remainder?
The question is, is there a theorem whose proposition is \[ f(h) = \sum_{m=0}^k \frac{f^{(m)}(0)h^m}{m!} + \mathcal O(h^{k+\delta}) \] with some $\delta\in(1,0)$. If so, what are sufficient assumptions? Trivially, my proposition holds if $f$ is $k+1$ times differentiable, but I want a slightly weaker assumption.