Is $2^{1+\sqrt{mn}}$ an upper bound for the size of the DFA resulting from the determinisation of n DFAs of size m (all automata sizes are expressed in number of states in this question)

There exist NFAs of size $n$ for which the smallest DFA recognizing the same language is of size at least $2^n$.

Let $\Sigma$ be a vocabulary, and let's consider $n$ regular languages $L_i \subseteq \Sigma^*$ respectively recognized by $n$ DFAs $D_i=\{Q_i, q^0_i, F_i , \delta_i \}$ whose size are bounded by $m$.

We consider the epsilon-free NFA $N$ built by merging the $n$ DFAs such as $N=\{Q=\bigcup Q_i, q^0, F=\bigcup F_i , \delta \}$ where $\delta = \bigcup \delta_i$ with the additional constraint that $\forall \sigma \in \Sigma \, \delta(q^0, \sigma) = \{q |  \delta_i(q_i^0, \sigma)=q \,  \, 0 \leq i \leq n \}$ (note that in $N$ the $q^0_i$ are not anymore initial states since there is a unique initial state $q^0$).

Let $D_N$ be the DFA resulting from the powerset construction of $N$ and $D_N^M$ be the DFA resulting from the minimisation of $D_N$.

Is the size of $D_N^M$ at most $2^{1+\sqrt{mn}}$ ?

An equivalent question is : let $L \subseteq \Sigma^*$ be the language resulting from the union of the $n$ regular languages $L_i$, is the size of the smallest DFA recognizing $L$ always below $2^{1+\sqrt{mn}}$ ?

I would appreciate if possible, some papers related to this problem or a proof/counter-example.

Many thanks, Luz
No.  Asymptotically, the size can be as large as $\sim m^n$.

Let $p_1,\dots,p_n$ denote the $n$ largest prime numbers that are at most $m$.  Let

$$L_i = \{x : x \not\equiv 0 \pmod{p_i}\},$$

where we assume the number $x$ is expressed in unary.  Then $L = L_1 \cup \dots \cup L_n$ has the form

$$L = \{x : x \not\equiv 0 \pmod q\},$$

where $q = p_1 \times \dots \times p_n$.

Note that, with this construction, each $L_i$ has a DFA of size at most $m$ (in particular, of size $p_i$).  However, the minimal DFA for $L$ has size $q$, which is close to $m^n$.  In particular, the size of the minimal DFA for $L$ is asymptotically larger than $2^{1 + \sqrt{mn}}$ (as a function of $n$, treating $m$ as fixed).