Sub exponentially hard OWF , PRF and iO I'm currently reading the work "Obfuscation of probabilistic circuits and Applications' by Canetti Lin Tessaro and Vaikuntanathan 2015. It says sub exponentially hard OWF implies 
sub exponentially hard PRF ( puncturable PRF) and later again proves that sub exponentially hard PRF ( puncturable PRF) + sub exponentially hard iO $\implies$ pIO(probabilistic circuits). 

I'm really confused with what the author means by the term "subexponentially hard" and is it a weather or a stronger assumption when it comes to the OWF or iO.

Thanks in advance
An efficiently computable function $f:\{0,1\}^*\rightarrow\{0,1\}^*$ is said to be $(s,\epsilon)$-one-way if for every adversary $\mathsf{A}$ of size* at most $s=s(|x|)$ the probability $\Pr\left[{\mathsf{A}(f(x))\in f^{-1}(x)}\right]$ is at most $\epsilon=\epsilon(|x|)$, where the probability is over uniform distribution on the domain and the random coins of $\mathsf{A}$. 

The standard one-wayness assumption is that $s=poly(|x|)$ and $\epsilon=negl(|x|)$ . A one-way function is subexponentially-hard if for a fixed constant $0&lt;c&lt;1$, it is $(2^{{|x|}^c},2^{-{|x|}^c})$-one-way. Note that the latter implies the former and is, therefore, a stronger assumption. 

Similar definitions apply for the other primitives too.

*Here, size refers either to the run-time if $\mathsf{A}$ is a probabilistic Turing machine, or the circuit-size in case $\mathsf{A}$ is a circuit.