<p>In this paper we study a new kind of geometric mean of positive definite operators <i>A</i> and <i>B</i>, named the near-geometric mean, of the form for <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(t\in [0,1]\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>t</mi> <mo>∈</mo> <mo stretchy="false">[</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">]</mo> </mrow> </math></EquationSource> </InlineEquation><Equation ID="Equ10"> <EquationSource Format="TEX">\(\begin{aligned} A \star _{t} B = A^{1/2} (B^{1/2} A^{-1} B^{1/2})^{t} A^{1/2}, \end{aligned}\)</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mtable> <mtr> <mtd columnalign="right"> <mrow> <mi>A</mi> <msub> <mo>⋆</mo> <mi>t</mi> </msub> <mi>B</mi> <mo>=</mo> <msup> <mi>A</mi> <mrow> <mn>1</mn> <mo stretchy="false">/</mo> <mn>2</mn> </mrow> </msup> <msup> <mrow> <mo stretchy="false">(</mo> <msup> <mi>B</mi> <mrow> <mn>1</mn> <mo stretchy="false">/</mo> <mn>2</mn> </mrow> </msup> <msup> <mi>A</mi> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </msup> <msup> <mi>B</mi> <mrow> <mn>1</mn> <mo stretchy="false">/</mo> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> </mrow> <mi>t</mi> </msup> <msup> <mi>A</mi> <mrow> <mn>1</mn> <mo stretchy="false">/</mo> <mn>2</mn> </mrow> </msup> <mo>,</mo> </mrow> </mtd> </mtr> </mtable> </mrow> </math></EquationSource> </Equation>arising from the determinantal inequality in diffusion tensor imaging. We show the geodesic property of near-geometric mean for the Thompson metric, the monotonicity on parameters and the boundedness of near-geometric mean. Moreover, we compare other known geometric means and Rényi mean with near-geometric mean in terms of the log-majorization.</p>