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\begin{aligned} min_{w_{1}} (δ) & = {\begin{cases} 0, & when w_{1} = 0 or 1, if (μ_{d i f f} - μ_{0}) \geq 0 \\ \frac{1}{2} N^{\frac{1}{2}} \frac{(μ_{d i f f} - μ_{0})}{σ}, & when w_{1} = \frac{1}{2}, if (μ_{d i f f} - μ_{0}) < 0 \end{cases} \\ max_{w_{1}} (δ) & = {\begin{cases} 0, & when w_{1} = 0 or 1, if (μ_{d i f f} - μ_{0}) < 0 \\ \frac{1}{2} N^{\frac{1}{2}} \frac{(μ_{d i f f} - μ_{0})}{σ}, & when w_{1} = \frac{1}{2}, if (μ_{d i f f} - μ_{0}) \geq 0 \end{cases} \end{aligned}

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\begin{align*} \min _{w_1} (\delta ) & = \left\{ \begin{array}{ll} 0, & \mbox{when} \quad w_1 = 0 \quad \mbox{or} \quad 1, \quad \mbox{if} \quad (\mu _\mr {diff} - \mu _0) \ge 0 \\ \frac{1}{2}N^\frac {1}{2} \frac{(\mu _\mr {diff} - \mu _0)}{\sigma }, & \mbox{when} \quad w_1 = \frac{1}{2}, \quad \mbox{if} \quad (\mu _\mr {diff} - \mu _0) < 0 \\ \end{array} \right. \\ \max _{w_1} (\delta ) & = \left\{ \begin{array}{ll} 0, & \mbox{when} \quad w_1 = 0 \quad \mbox{or} \quad 1, \quad \mbox{if} \quad (\mu _\mr {diff} - \mu _0) < 0 \\ \frac{1}{2}N^\frac {1}{2} \frac{(\mu _\mr {diff} - \mu _0)}{\sigma }, & \mbox{when} \quad w_1 = \frac{1}{2}, \quad \mbox{if} \quad (\mu _\mr {diff} - \mu _0) \ge 0 \\ \end{array} \right. \\ \end{align*}