<p>Let <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(\mathcal {S}_{T}^{*}\)</EquationSource> <EquationSource Format="MATHML"><math> <mmultiscripts> <mi mathvariant="script">S</mi> <mrow> <mi>T</mi> </mrow> <mrow> <mrow /> <mo>∗</mo> </mrow> </mmultiscripts> </math></EquationSource> </InlineEquation> denote the class of normalized analytic functions <i>f</i> such that <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(zf^{\prime }(z)\diagup f(z)\prec e^{z+\frac{z^{2}}{2}}\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>z</mi> <msup> <mi>f</mi> <mo>′</mo> </msup> <mrow> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> </mrow> <mi>∕</mi> <mi>f</mi> <mrow> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> </mrow> <mo>≺</mo> <msup> <mi>e</mi> <mrow> <mi>z</mi> <mo>+</mo> <mfrac> <msup> <mi>z</mi> <mn>2</mn> </msup> <mn>2</mn> </mfrac> </mrow> </msup> </mrow> </math></EquationSource> </InlineEquation> for <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(\left| z\right| &lt;1\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mfenced close="|" open="|"> <mi>z</mi> </mfenced> <mo>&lt;</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation>. The structural formula, inclusion relations, coefficient estimates and various radii constants for functions in the class <InlineEquation ID="IEq4"> <EquationSource Format="TEX">\(\mathcal {S}_{T}^{*}\)</EquationSource> <EquationSource Format="MATHML"><math> <mmultiscripts> <mi mathvariant="script">S</mi> <mrow> <mi>T</mi> </mrow> <mrow> <mrow /> <mo>∗</mo> </mrow> </mmultiscripts> </math></EquationSource> </InlineEquation> are obtained.</p>