We constrain some lepton flavor violating vertices in the context of the two Higgs doublet model type III, assuming that the lightest scalar Higgs mass $% m_{h^{0}} $is about $115 $GeV. Specifically, based on the $g-2 $muon factor and the decay width of$ \mu \to e\gamma $,$ $the following quite general bounds are obtained:$ 7.62\times 10^{-4}\lesssim \xi_{\mu \tau}^{2}\lesssim 4.44\times 10^{-2}, $ $\xi_{e\tau}^{2}\lesssim 2.77\times 10^{-14}$. Additionally, based on the processes $\tau \to \mu \gamma $, and $\tau \to \mu \mu \mu $, bounds on $\xi_{\tau \tau} $and $\xi_{\mu \mu} $are also gotten, such constraints on these parameters$ $still give enough room for either a strong suppression$ $or strong enhancement on the coupling of any Higgs boson to a pair of tau leptons or a pair of muons\textbf{.} Furthermore, upper limits on the decay widths of the leptonic decays $\tau \to e\gamma, $and $\tau \to eee $are calculated, finding them to stay far from the reach of near future experiments.