  We provide generalizations of a class of stochastic Gronwall inequalities
that has been studied by von Renesse and Scheutzow (2010), Scheutzow (2013),
Xie and Zhang (2020) and Mehri and Scheutzow (2021). This class of stochastic
Gronwall inequalities is a useful tool for SDEs.
  Our focus are convex generalizations of the Bihari-LaSalle type. The
constants we obtain are sharp. In particular, we provide new sharp constants
for the stochastic Gronwall inequalities. The proofs are connected to a
domination inequality by Lenglart (1977), an inequality by Pratelli (1976) and
a characterization of Lenglart's concept of domination via the Snell envelope.
  The inequalities we study appear for example in connection with exponential
moments of solutions to path-dependent SDEs: For non-path-dependent SDEs,
criteria for the finiteness of exponential moments are known. To be able to
extend these proofs to the path-dependent case, a convex generalization of a
stochastic Gronwall inequality seems necessary. Using the results of this
paper, we obtain a criterion for the finiteness of exponential moments which is
similar to that known for non-path-dependent SDEs.
  Stochastic Gronwall inequalities can also be applied to study other types of
SDEs than path-dependent SDEs: An estimate of this paper is applied by Agresti
and Veraar (2023) to prove global well-posedness for reaction-diffusion systems
with transport noise.
