lower equal with dot

\documentclass{article}
\usepackage{amssymb,stackengine}
\newcommand\ledot{\mathrel{\ensurestackMath{%
  \stackengine{-.5ex}{\lessdot}{-}{U}{c}{F}{F}{S}}}}
\begin{document}
$a\ledot b$
\end{document}

If you need it across mathstyles:

\documentclass{article}
\usepackage{amssymb,stackengine,scalerel}
\newcommand\ledot{\mathrel{\ThisStyle{\ensurestackMath{%
  \stackengine{-.5\LMex}{\SavedStyle\lessdot}{\SavedStyle-}{U}{c}{F}{F}{S}}}}}
\begin{document}
$a\ledot b$

$\scriptstyle a\ledot b$

$\scriptscriptstyle a\ledot b$
\end{document}

ADDENDUM

Comments from the OP indicate a desire to add a subscript to the symbol. The above formulations have too much depth to make that work well directly, so here is an alternative in which the underset is \smashed and the underset depth is adjusted a bit:

\documentclass{article}
\usepackage{amssymb,stackengine,scalerel}
\newcommand\ledot{\mathrel{\ThisStyle{\ensurestackMath{%
  \stackengine{.8\LMex}{\SavedStyle\lessdot}{\smash{\SavedStyle-}}%
  {U}{c}{F}{F}{S}}}}}
\begin{document}
$a\ledot_{\!P} b$

$\scriptstyle a\ledot_{\!P} b$

$\scriptscriptstyle a\ledot_{\!P} b$
\end{document}

There is also a specific symbol \leqslantdot with MnSymbol package, or fdsymbol.

Edit: We observe that the \leqslantdot with MnSymbol not changes the CM fonts.

\documentclass[12pt]{article}
\usepackage{amsmath}
\usepackage{MnSymbol}
\begin{document}
$f(a,b)\leqslantdot g(b,a)$
\end{document}

---

While if you use fdsymbol, you have not an exact dot but it is similar to the square with a zoom and there is a change of the font of the rounded brackets (for example).

\documentclass[12pt]{article}
\usepackage{amsmath}
\usepackage{fdsymbol}
\begin{document}
$f(a,b)\leqslantdot g(b,a)$
\end{document}