$\lim _{n\to \infty }F_{1,n}(z),\qquad \lim _{n\to \infty }G_{1$Although the title of this article specifies analytic functions, there are results for more general functions of a complex variable as well.

There are several notations describing infinite compositions, including the following:

**Forward compositions:** *F*_{k,n}(*z*) = *f*_{k} ∘ *f*_{k+1} ∘ ... ∘ *f*_{n−1} ∘ *f*_{n}(*z*).

**Backward compositions:** *G*_{k,n}(*z*) = *f*_{n} ∘ *f*_{n−1} ∘ ... ∘ *f*_{k+1} ∘ *f*_{k}(*z*)

In each case convergence is interpreted as the existence of the following limits:

- $\lim _{n\to \infty }F_{1,n}(z),\qquad \lim _{n\to \infty }G_{1,n}(z).$

For convenience, set *F*_{n}(*z*) = *F*_{1,n}(*z*) and *G*_{n}(*z*) = *G*_{1,n}(*z*).

One may also write $F}_{n}(z)=\underset{}{\stackrel{}{}$

**Forward compositions:** *F*_{k,n}(*z*) = *f*_{k} ∘ *f*_{k+1} ∘ ... ∘ *f*_{n−1} ∘ *f*_{n}(*z*).

**Backward compositions:** *G*_{k,n}(*z*) = *f*_{n} ∘ *f*_{n−1} ∘ ... ∘ *f*_{k+1} ∘ *f*_{k}(*z*)

In each case convergence is interpreted as the existence of the following limits:

For convenience, set *F*_{n}(*z*) = *F*_{1,n}(*z*) and *G*_{n}(*z*) = *G*_{1,n}(*z*).

One may also write $F_{n}(z)={\underset {k=1}{\overset {$

One may also write $F_{n}(z)={\underset {k=1}{\overset {n}{\mathop {R} }}}\,f_{k}(z)=f_{1}\circ f_{2}\circ \cdots \circ f_{n}(z)$ and
$G_{n}(z)={\underset {k=1}{\overset {n}{\mathop {L} }}}\,g_{k}(z)=g_{n}\circ g_{n-1}\circ \cdots \circ g_{1}(z)$

Many results can be considered extensions of the following result:

**Contraction Theorem for Analytic Functions.**^{[1]} Let *f* be analytic in a simply-connected region *S* and continuous on the closure *S* of *S*. Suppose *f*(*S<*Let {*f*_{n}} be a sequence of functions analytic on a simply-connected domain *S*. Suppose there exists a compact set Ω ⊂ *S* such that for each *n*, *f*_{n}(*S*) ⊂ Ω.

**Forward (inner or right) Compositions Theorem.** {*F*_{n}} converges uniformly on compact subsets of *S* to a constant function *F*(*z*) = λ.^{[2]}

**Backward (outer or left) Compositions Theorem.** {*G*_{n}} converges uniformly on compact subsets of *S* to γ ∈ Ω if and only if the sequence of fixed points {*γ*_{n}} of the {*f*_{n}} converges to *γ*.^{[3]}

Additional theory resulting from investigations based on these two theorems, particularly Forward Compositions Theorem, include location analysis for the limits obtained here [1]. For a different approach to Backward Compositions Theorem, see [2].

Regarding Backward Compositions Theorem, the example *f*_{2n}(*z*) = 1/2 and *f*_{2n−1}(*z*) = −1/2 for *S* = {*z* : |*z*| < 1} demonstrates the inadequacy of simply requiring contraction into a compact subset, like Forward Compositions Theorem.

For functions not necessarily analytic the Lipschitz condition suffices:

**Theorem.**^{[4]} Suppose $S$ is a simply connected compact subset of $$Regarding Backward Compositions Theorem, the example *f*_{2n}(*z*) = 1/2 and *f*_{2n−1}(*z*) = −1/2 for *S* = {*z* : |*z*| < 1} demonstrates the inadequacy of simply requiring contraction into a compact subset, like Forward Compositions Theorem.

For functions not necessarily analytic the Lipschitz condition suffices:

**Theorem.**^{[4]} Suppose Lipschitz condition suffices:
Results^{[5]} involving **entire functions** include the following, as examples. Set

- ${\begin{aligned}f_{n}(z)&=a_{n}z+c_{n,2}z^{2}+c_{n,3}z^{3}+\cdots \\\rho _{n}&=\sup _{r}\left\{\left|c_{n,r}\right|^{\frac {1}{r-1}}\right\}\end{aligned}}$

Then the following results hold:

**Theorem E1.**^{[6]} If *a*_{n} ≡ 1,
- $\sum _{n=1}^{\infty }\rho _{n}<\infty$

- then
*F*_{n} → *F*, entire.

**Theorem E2.**^{[5]} Set ε_{n} = |*a*_{n}−1| suppose there exists non-negative δ_{n}, *M*_{1}, *M*_{2}, *R* such that the following holds:
- ${\begin{aligned}\sum _{n=1}^{\infty }\varepsilon _{n}&<\infty ,\\\sum _{n=1}^{\infty }\delta _{n}&<\infty ,\\\prod _{n=1}^{\infty }(1+\delta _{n})&<M_{1},\\\prod _{n=1}^{\infty }(1+\varepsilon _{n})&<M_{2},\\\rho _{n}&<{\frac {\delta _{n}}{RM_{1}M_{2}}}.\end{aligned}}$

- Then
*G*_{n}(*z*) → *G*(*z*), analytic for |*z*| < *R*. Convergence is uniform on compact subsets of {*z* : |*z*| < *R*}.

**Additional elementary results include:**

**Theorem GF3.**^{[4]} Suppose $f_{n}(z)=z+\rho _{n}\varphi (z)$ where there exist $R,M>0$ such that $|z|<R$ implies $|\varphi (z)|<M.$ FurthermoThen the following results hold:

**Theorem E1.**^{[6]} If *a*_{n} ≡ 1,
- $\sum _{n=1}^{\infty }\rho _{n}<\infty$

- then
*F*_{n} → *F*, entire.

**Theorem E2.**^{[5]} Set ε_{n} = |*a*_{n}−1| suppose there exists non-negative δ_{n}, *M*_{1}, *M*_{2}, *R* such that the following holds:
- $f_{n}(z)=z+\rho _{n}\varphi (z)$