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Let f : (Cn+1, 0) → (C, 0) be a holomorphic germ defining an isolated hypersurface singularity V at the origin. Let μ and ν and pg be the Milnor number, multiplicity and geometric genus of (V, 0), respectively. We conjecture that μ ≥ (ν − 1)n+1 and the equality holds if and only if f is a semi-homogeneous function. We prove that this inequality holds for n = 1, and also for n = 2 or 3 with additional assumption that f is a quasihomogeneous function. For n = 1, if V has at most two irreducible branches at the origin, or if f is a quasi-homogeneous function, then μ = (ν − 1)2 if and only if f is a homogeneous polynomial. For n = 2, if f is a quasihomogeneous function, then μ = (ν − 1)3 iff 6pg = ν(ν − 1)(ν − 2) iff f is a homogeneous polynomial after biholomorphic change of variables. For n = 3, if f is a quasi-homogeneous function, then μ = (ν − 1)4 iff 24pg = ν(ν − 1)(ν − 2)(ν − 3) iff f is a homogeneous polynomial after biholomorphic change of variables.

A remark on lower bound of Milnor number and characterization of homogeneous hypersurface singularities