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Abstract This paper focuses on the minimum eigenvalue involving the Fan product. By utilizing the Hölder inequality and the classic eigenvalue inclusion theorem, we introduce two novel lower bounds for τ ( A 1 ⋆ A 2 ) $\tau \left ( A_{1}\star A_{2} \right )$ , representing the minimum eigenvalue involving the Fan product of two M-matrices A 1 , A 2 $A_{1},A_{2}$ . The newly derived lower bounds are then compared with the traditional findings. Numerical tests are presented to illustrate that the new lower bound formulas significantly enhance Johnson and Horn’s results in certain scenarios and are more precise than other existing findings.