= Solution
Let the incoming projectile momentum be $\mathbf p_1$. The stationary target has <four-momentum> $(m_2c,\mathbf0)$, so the total incoming norm is
$$
P_{\rm in}^2=\left(\frac{E_1}{c}+m_2c\right)^2-|\mathbf p_1|^2=m_1^2c^2+m_2^2c^2+2m_2E_1.
$$
Here the <relativistic energy-momentum relation> eliminated $|\mathbf p_1|^2$. Write $W$ for the total energy in the <centre-of-momentum frame>; then $W^2=c^2P_{\rm in}^2$. If $M_f=\sum_i\mu_i$, the <relativistic collision mass threshold> gives $W\geq M_fc^2$. Squaring and rearranging with $m_2>0$ yields the <fixed-target production threshold>
$$
\boxed{E_1\geq\frac{M_f^2-m_1^2-m_2^2}{2m_2}c^2.}
$$
The actual incident total energy also obeys $E_1\geq m_1c^2$. Thus a negative algebraic reaction bound would not allow a negative-energy projectile.
For a <photon>, $m_1=0$ and $E_1=h\nu$, where $h$ is the <Planck constant>. The necessary bound becomes
$$
\nu\geq\frac{M_f^2-m_2^2}{2m_2h}c^2.
$$
In the usual massive multi-product channel with $M_f>m_2$, conservation alone permits the threshold configuration in which all products are at rest relative to one another in the <centre-of-momentum frame>. Boosting that configuration supplies the required laboratory momentum. Hence the <photon fixed-target production threshold> is
$$
\boxed{\nu_{\min}=\frac{(\sum_i\mu_i)^2-m_2^2}{2m_2h}c^2\qquad(M_f>m_2).}
$$
For a multi-product channel with $M_f\leq m_2$, there is no positive mass threshold: the lower limit of allowed positive frequencies is zero. The compact expression for that lower limit is $\boxed{\nu_{\rm inf}=\max\{0,M_f^2-m_2^2\}c^2/(2m_2h)}$; it need not correspond to a zero-frequency incident photon.
For completeness, if the only product is one particle of mass $\mu_1$, its invariant mass must match exactly. A positive incident frequency is possible only when $\mu_1>m_2$, and is then fixed at $\nu=(\mu_1^2-m_2^2)c^2/(2m_2h)$; frequencies above that value do not produce the same single particle while conserving four-momentum. If a multi-product channel includes massless particles, an all-products-at-rest configuration is impossible for those particles with nonzero energy, so the nominal massive threshold may be only an infimum. These qualifications distinguish the requested necessary inequalities from unrestricted claims of sufficiency.
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