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We analyze the prospect for observing the lightest and heaviest CP-evenneutral Higgs bosons ($\phi=h,\;H$) in their decays to flavor violating$b\bar{s}$ (with charge conjugate) final states at the future Large Hadronelectron Collider (LHeC), in the framework of the Two Higgs Doublet Model typeIII (2HDM-III) assuming a four-zero texture in the Yukawa matrices and ageneral Higgs potential. We consider the charge current production processes$e^- p \to \nu_e \phi q_f$ ($\phi= h, \; H$), with the flavor violating decaysof the Higgs bosons. We consider scenarios of the model which are consistentwith current experimental data from flavor and Higgs physics. We also considerthe most relevant Standard Model (SM) backgrounds. We find that the lightest(SM-like) CP-even Higgs boson $h$ and the heaviest one $ H$ (with mass up to150 GeV) would have 1--2 $\Sigma$ and 1 $\Sigma$ significances, respectively,with 100 fb$^{-1}$ of data. At the end of the LHeC running, the significanceswould be increased by ten times and the detection of both Higgs bosons ($h, H$)would be guaranteed.

Flavor violating signatures of neutral Higgs bosons at the LHeC