# Partition complexes and trees

Gijs Heuts  and  Ieke Moerdijk

###### Abstract.

We construct a homotopy initial functor from the partition complex of a finite
set $A$ to a category of trees with leaves labelled by $A$. As an application,
this provides an equivalence between different bar constructions of an operad.
In the differential graded case, this gives a very elementary proof of an
equivalence originally due to Fresse.

##  1\. Introduction

Let $A$ be a finite set. The partitions of $A$ form a partially ordered set
under refinement. This poset has an initial object (the indiscrete partition)
and a final object (the discrete partition); removing these two extreme
partitions leaves a poset for which we write $\mathbf{P}(A)$. The nerve
$N\mathbf{P}(A)$ is a simplicial set called the _partition complex_ of $A$.
These partition complexes play a central role in the study of bar-cobar (or
Koszul) duality for operads [4, 2], as we will review in Section 4.

In this short note we give a precise and elementary relation between the
partition complex $N\mathbf{P}(A)$ and a certain category $\mathbf{T}(A)$ of
rooted trees with set of leaves $A$. This category has a morphism $f\colon
T\rightarrow S$ if the tree $T$ can be obtained from $S$ by contracting a set
of inner edges of $S$. The category $\mathbf{T}(A)$ has an initial object,
namely the ‘corolla’ $C_{A}$ with leaves $A$ and no internal edges. Removing
this object from $\mathbf{T}(A)$ gives a subcategory that we denote
$\mathbf{T}(A)^{+}$.

We write $\mathbf{\Delta}/N\mathbf{P}(A)$ for the category of simplices of the
simplicial set $N\mathbf{P}(A)$, which can be thought of as a category of
layered trees. It has the same weak homotopy type as $N\mathbf{P}(A)$ itself
(see Section 2), and our result is as follows:

######  Theorem 1.1.

There is a functor

$\varphi\colon\mathbf{\Delta}/N\mathbf{P}(A)\rightarrow\mathbf{T}^{+}(A)$

with the property that for every object $T$ of $\mathbf{T}^{+}(A)$, the slice
category $\varphi/T$ is weakly contractible. In particular, $\varphi$ induces
a weak homotopy equivalence between the classifying spaces of these two
categories.

The second statement of the theorem is a consequence of the first by Quillen’s
Theorem A [10]. However, the first statement is significantly sharper than
just $\varphi$ inducing a homotopy equivalence of classifying spaces. A
functor $\varphi\colon\mathbf{D}\rightarrow\mathbf{C}$ with the property that
$\varphi/c$ is a weakly contractible category for each object $c$ of
$\mathbf{C}$ is called _homotopy initial_. The terminology refers to the fact
that precomposing by such a functor $\varphi$ preserves homotopy limits.

The essential content of this paper consists of Sections 2 and 3, proving
Theorem 1.1. We will see that our result implies Fresse’s comparison of
different versions of the bar construction [4, 8] for a differential graded
operad (see also [13]). In Section 4 we illustrate some of the uses of the
category $\mathbf{T}(A)$ of trees and in particular see how it is related to
Ching’s description of the bar construction of a topological operad [2]. In
the appendix we explain some of the properties of homotopy initial functors.

##  2\. Partition complexes and layered trees

Fix a finite set $A$. We introduced its partition complex $N\mathbf{P}(A)$
above. The $p$-simplices of this simplicial set are chains of length $p$ of
nontrivial partitions, which can be thought of as _layered trees_ with $p$
internal layers. For example, the partitions

$(abcde)(f)\leq(ab)(cde)(f)$

of the set $A=\\{a,b,c,d,e,f\\}$ define a $1$-simplex in $N\mathbf{P}(A)$
represented by the layered tree

$1$$0$$a$$b$$c$$d$$e$$f$

Here the layer labelled 1 represents the partition $(ab)(cde)(f)$, whereas the
layer labelled 0 represents $(abcde)(f)$. Let us fix some terminology.

(1) A simplex in $N\mathbf{P}(A)$ is nondegenerate if it has no layer at which
all vertices are unary (meaning they have only one incoming edge, if we orient
our trees towards the root). A simplex is called _elementary_ if it is has
exactly one non-unary vertex at each layer. As an example, observe that the
picture above does not represent an elementary simplex, but the trees

$2$$1$$0$$a$$b$$c$$d$$e$$f$$2$$1$$0$$a$$b$$c$$d$$e$$f$

do represent elementary simplices, namely the two chains of partitions

$\displaystyle(abcde)(f)\leq(ab)(cde)(f)\leq(a)(b)(cde)(f),$
$\displaystyle(abcde)(f)\leq(ab)(cde)(f)\leq(ab)(c)(d)(e)(f).$

As this example illustrates, any nondegenerate simplex in $N\mathbf{P}(A)$ is
a face of an elementary one, but the latter need not be unique.

(2) The category of simplices of $N\mathbf{P}(A)$ is denoted
$\mathbf{\Delta}/N\mathbf{P}(A)$. Objects of this category are _layered trees_
(with $A$ as set of leaves), morphisms are compositions of face operations
deleting a layer and contracting all edges in that layer, and degeneracy
operations inserting a new layer consisting entirely of unary vertices. There
is a functor

$\zeta\colon\mathbf{\Delta}/N\mathbf{P}(A)\longrightarrow\mathbf{P}(A)$

sending a chain of partitions to its final element. This functor induces a
homotopy equivalence of classifying spaces. Indeed, as is well known, this is
the case for any small category $\mathbf{C}$ in place of $\mathbf{P}(A)$,
since the projection

$\zeta\colon\mathbf{\Delta}/N\mathbf{C}\longrightarrow\mathbf{C}$

sending $(c_{0}\rightarrow\cdots\rightarrow c_{p})$ to $c_{p}$ satisfies the
hypothesis of Quillen’s Theorem A again, see [12].

##  3\. Trees without layers and the main theorem

There is a more basic category of trees $\mathbf{T}(A)$ whose objects are
isomorphism classes of finited rooted trees with $A$ as set of leaves, in
which every vertex has at least two incoming edges. Morphisms in this category
are compositions of ‘face maps’ creating an inner edge, as in

$\partial_{f}$$e$$e$$f$

Each such morphism $S\rightarrow T$ maps edges to edges, while it preserves
the root and induces the identity on the set of leaves. Note that a morphism
$S\rightarrow T$ exists only if $S$ can be obtained from $T$ by contracting a
sequence of inner edges and is unique in that case. Thus $\mathbf{T}(A)$ is in
fact a poset. This category of trees is the same as the one considered by
Ching [2] and by Hoffbeck and the second author [7].

The category $\mathbf{T}(A)$ has an initial object, namely the minimal tree
with leaves $A$ connected by a single vertex to its root: we label this tree
$C_{A}$ and refer to it as the corolla with $A$ leaves. We shall write
$\mathbf{T}^{+}(A)$ for the full subcategory of $\mathbf{T}(A)$ obtained by
omitting this initial object; in other words, the full subcategory of trees
with at least one inner edge.

Forgetting the layers and deleting unary vertices now defines a functor from
layered trees to trees,

$\varphi\colon\mathbf{\Delta}/N\mathbf{P}(A)\rightarrow\mathbf{T}^{+}(A),$

which is the functor of Theorem 1.1. The remainder of this section is devoted
to its proof.

###### Proof of Theorem 1.1.

For an object $T$ of $\mathbf{T}^{+}(A)$, a _layering_ of $T$ is a $p$-simplex
$\sigma$ of $N\mathbf{P}(A)$ for which $\varphi(\sigma)=T$. An _elementary_
such layering corresponds to a linear ordering of the vertices of $T$
compatible with the partial ordering defined by the tree structure. In
particular, $p=|V(T)|-2$ in this case, where $V(T)$ is the set of vertices of
$T$. If $\tau$ is a face of a layering of $T$, then there exists a map
$\varphi(\tau)\rightarrow T$ in $\mathbf{T}^{+}(A)$. Vice versa, if $\sigma$
is a level tree for which $\varphi(\sigma)$ maps to $T$, then $\sigma$ is a
face of a layering of $T$.

Now fix a tree $T$ in $\mathbf{T}^{+}(A)$. The slice $\varphi/T$ is the
category of simplices of a simplicial set $L(T)$ generated by the
$(|V(T)|-2)$-simplices representing elementary layerings of $T$. The corolla
$C_{A}$ is not an object of $\mathbf{T}^{+}(A)$, but we introduce the
convention that $L(C_{A})=\varnothing$. Also, in the course of the argument
below, we will consider trees in $\mathbf{T}^{+}(B)$ for sets $B$ different
from $A$.

We call a vertex $v$ of $T$ a _leaf vertex_ if all its incoming edges are
leaves of the tree $T$. For such a $v$, let $L^{v}(T)\subseteq L(T)$ be the
simplicial subset generated by simplices which represent elementary layerings
of $T$ with $v$ on top. Then

$L(T)=\bigcup_{v}L^{v}(T)$

with $v$ ranging over leaf vertices of $T$. Moreover, for any such $v$ the
simplicial set $L^{v}(T)$ is clearly a cone on the simplicial set
$L(\partial_{v}T)$, with $\partial_{v}T$ the tree obtained from $T$ by
removing the vertex $v$ and its incoming edges. To be precise, $L^{v}(T)$ may
be identified with the join $L(\partial_{v}T)\star\Delta[0]$. (This
identification results from the observation that any layering of
$\partial_{v}T$, or a face of it, may be naturally upgraded to an object of
$L^{v}(T)$ by attaching the corolla with vertex $v$ at the top.) Similarly, if
$v_{1},\ldots,v_{k}$ is a finite set of leaf vertices of $T$, then the
intersection

$L^{v_{1}}(T)\cap\cdots\cap L^{v_{n}}(T)$

is a cone on $L(\partial_{v_{1},\ldots,v_{n}}T)$, where
$\partial_{v_{1},\ldots,v_{n}}T$ is obtained from $T$ by removing all of the
vertices $v_{1},\ldots,v_{n}$ and their incoming edges. Thus, the $L^{v}(T)$
together form a cover of $L(T)$ by weakly contractible simplicial sets, all of
whose finite intersections are again contractible. So $L(T)$ is itself weakly
contractible, which proves the theorem. ∎

######  Example 3.1.

An operad $\mathbf{O}$ in the category of sets defines a presheaf
$N\mathbf{O}$ on $\mathbf{T}^{+}(A)$ called its _nerve_. This presheaf assigns
to any tree $T$ the set of labellings of the vertices of $T$ by operations in
$\mathbf{O}$ of arity corresponding to the number of inputs of the vertex. The
functoriality with respect to edge contractions uses the composition of
operations in $\mathbf{O}$. The category of elements of $N\mathbf{O}$ is
denoted

$\mathbf{T}^{+}(A,\mathbf{O}):=\mathbf{T}^{+}(A)/N\mathbf{O}.$

The presheaf $N\mathbf{O}$ pulls back to a presheaf $\varphi^{*}(N\mathbf{O})$
on $\mathbf{\Delta}/N\mathbf{P}(A)$ whose category of elements we denote
$\mathbf{\Delta}/N\mathbf{P}(A)_{\mathbf{O}}$. It is the category of elements
of a simplicial set $N\mathbf{P}(A)_{\mathbf{O}}$. This simplicial set is a
_labelled partition complex_ of the kind considered in [4]. As explained in
the appendix, our theorem implies that the functor

$\mathbf{\Delta}/N\mathbf{P}(A)_{\mathbf{O}}\longrightarrow\mathbf{T}^{+}(A,\mathbf{O})$

is again homotopy initial, so in particular a weak homotopy equivalence. The
same applies if $\mathbf{O}$ is an operad in simplicial sets, or in chain
complexes, for example. (As explained in the appendix, really any symmetric
monoidal model category or $\infty$-category could be used instead.) This
gives a different and more direct proof of the quasi-isomorphism of Fresse
between different bar constructions for an operad proved in [4] and mentioned
in the introduction, see also the discussion in the next section. Note that
Fresse’s proof goes via a construction in the opposite direction, from trees
to layered trees, which he calls ‘levelization’.

##  4\. More on the category of trees

In this concluding section we collect some remarks on the category of trees
$\mathbf{T}(A)$ described above, which should illustrate its relevance and
some of its basic properties.

(1) As $A$ ranges over finite sets, the categories $\mathbf{T}(A)$ (as well as
$\mathbf{T}^{+}(A)$) form an operad in the category of small categories.
Indeed, for $a\in A$, a composition map of the form

$\mathbf{T}(B)\circ_{a}\mathbf{T}(A)\rightarrow\mathbf{T}(A\circ_{a}B)$

is given by grafting trees with leaves $B$ onto the leaf $a$ of a tree with
leaves $A$. Here $A\circ_{a}B$ denotes the set $(A-\\{a\\})\amalg B$. An
algebra for this operad is a category with a colax symmetric monoidal
structure, as has been emphasized by Hinich.

(2) The homotopy type of the classifying space of $\mathbf{T}^{+}(A)$ (or
equivalently of the partition complex $N\mathbf{P}(A)$) is a wedge of
$(|A|-1)!$ spheres of dimension $|A|-3$ (cf. [11]).

(3) Let $\mathcal{E}$ be a symmetric monoidal simplicial model category (or
symmetric monoidal $\infty$-category) with a zero object. Then there is a
‘trivial’ operad $\mathbf{1}$ having the monoidal unit in arity 1 and the zero
object in all others. If $\mathbf{O}\rightarrow\mathbf{1}$ is an augmented
operad in $\mathcal{E}$ one may form its simplicial bar construction, with
object of $p$-simplices

$B(\mathbf{1},\mathbf{O},\mathbf{1})_{p}=\mathbf{O}\circ\cdots\circ\mathbf{O}.$

Here there are $p$ copies of $\mathbf{O}$ on the right, the symbol $\circ$
denotes the usual composition product of symmetric sequences, the inner face
maps use the operad structure of $\mathbf{O}$, and the outer face maps apply
the augmentation. The realization of this simplicial object in $\mathcal{E}$
is (one possible version of) the bar construction $B\mathbf{O}$ of the operad
$\mathbf{O}$.

It is not difficult to see (cf. [4, Section 4.3]) that this $B\mathbf{O}$ can
be described explicitly in terms of the labelled partition complexes
introduced above. Indeed, the term $\mathbf{O}^{\circ
p}(A)=B(\mathbf{1},\mathbf{O},\mathbf{1})_{p}(A)$ consists of layered trees of
height $p$ with leaves $A$, and each internal vertex labelled by an operation
of $\mathbf{O}$. Combining this with our result, one deduces fairly easily
that $B\mathbf{O}(A)$ is a suspension of the cofiber of the map

$\mathrm{hocolim}_{T\in\mathbf{T}^{+}(A)}N\mathbf{O}(T)\rightarrow\mathrm{hocolim}_{T\in\mathbf{T}(A)}N\mathbf{O}(T)\simeq\mathbf{O}(A).$

This gives an alternative description for the simplicial bar construction in
terms of trees. In the case where $\mathcal{E}$ is the category of chain
complexes over a commutative ring, this reproduces the comparison of bar
constructions given by Fresse [4]. If $\mathcal{E}$ is the category of
topological spaces or spectra, Ching [2] observes that the simplicial bar
construction is even _homeomorphic_ to the one defined in terms of trees; our
result is a bit weaker, but applies more generally.

(4) If again $\mathbf{O}$ is an operad in a suitable simplicial model
category, then the term $\mathrm{hocolim}_{T\in\mathbf{T}(A)}N\mathbf{O}$ is
essentially the Boardman–Vogt $W$-construction $W\mathbf{O}(A)$ (as described
in [1]). Indeed, the latter can be represented by ‘the space’ of trees with
leaves $A$, vertices labelled by operations of $\mathbf{O}$ and inner edges
labelled by a length $t$ in the 1-simplex $\Delta^{1}$. If such a length is 0,
then the resulting tree is identified with the one where that inner edge is
contracted and the operations corresponding to its endpoints are composed.
Viewed in this way, the subobject
$\mathrm{hocolim}_{T\in\mathbf{T}^{+}(A)}N\mathbf{O}(T)$ can be thought of as
the _decomposable_ operations in $W\mathbf{O}(A)$, i.e., the ones arising as a
composition of two or more operations in the operad $W\mathbf{O}$. Writing
$\mathrm{Indec}(W\mathbf{O}(A))$ for the quotient of $W\mathbf{O}(A)$ by the
decomposable operations, the previous remark implies an equivalence of
symmetric sequences

$B\mathbf{O}\simeq\Sigma\mathrm{Indec}(W\mathbf{O}).$

In the context of topological operads, this observation goes back to
Salvatore.

(5) The Fulton–MacPherson operad $\mathbf{FM}_{d}$ has as its terms
$\mathbf{FM}_{d}(A)$ certain manifolds with corners that compactify the
configuration spaces $\mathrm{Conf}_{d}(A)$ of $|A|$ points in
$\mathbb{R}^{d}$, or rather their quotients by translation and dilation (cf.
the discussion in [5, Section 1.8]). These manifolds are naturally stratified,
with strata indexed by the objects of the category $\mathbf{T}(A)$. The
stratum corresponding to a tree is homeomorphic to a product over the vertices
of $T$ of the configuration spaces corresponding to the inputs of each vertex;
the corolla $C_{A}$ corresponds to the maximal open stratum. The operad
$\mathbf{FM}_{d}$ is weakly equivalent to the operad $\mathbf{E}_{d}$ of
little $d$-cubes but has more favourable cofibrancy properties.

Using the description of $\mathbf{FM}_{d}(A)$ as a stratified space, it is not
difficult to see that the homotopy colimit

$\mathrm{hocolim}_{\mathbf{T}^{+}(A)}N\mathbf{FM}_{d}(-)$

is equivalent to the boundary of the manifold $\mathbf{FM}_{d}(A)$. Thus, one
deduces from (3) above that the terms of the bar construction (and hence of
the bar construction of the little $d$-disks operad $\mathbf{E}_{d}$) are
equivalent to the spaces

$\Sigma\bigl{(}\mathbf{FM}_{d}(A)/\partial\mathbf{FM}_{d}(A)\bigr{)}.$

The reader might compare this to the remark immediately following [3, Lemma
9.5]. This reference contains a much more detailed analysis of the bar
construction of the Fulton–MacPherson operad and includes a proof of the
‘self-duality’ of the $\mathbf{E}_{d}$-operad.

## Appendix

This appendix is a service to the reader not familiar with homotopy initial
functors, illustrating some useful properties of this notion. None of this
material is original. A standard account in the context of model categories is
[6, Section 19.6] or [9, Section 4.1] in the context of $\infty$-categories.

If $\varphi\colon\mathbf{D}\rightarrow\mathbf{C}$ is a functor and $c$ an
object of $\mathbf{C}$, then $\varphi/c$ (or simply $\mathbf{D}/c$, leaving
$\varphi$ implicit) denotes the category with objects the pairs consisting of
an object $d$ of $\mathbf{D}$ and a morphism
$\alpha\colon\varphi(d)\rightarrow c$, and with morphisms from $\alpha$ to
another $\alpha^{\prime}\colon\varphi(d^{\prime})\rightarrow c$ the morphisms
$\beta\colon d\rightarrow d^{\prime}$ in $\mathbf{D}$ making the resulting
triangle

${\varphi(d)}$${\varphi(d^{\prime})}$${c}$$\scriptstyle{\alpha}$$\scriptstyle{\varphi(\beta)}$$\scriptstyle{\alpha^{\prime}}$

commute. The functor $\varphi$ is called _homotopy initial_ (or _homotopy left
cofinal_ by some) if for each object $c$ of $\mathbf{C}$, the category
$\varphi/c$ has contractible classifying space.

Quillen’s Theorem A [10] states that if $\varphi$ is homotopy initial, then it
induces a homotopy equivalence of classifying spaces $B\mathbf{D}\simeq
B\mathbf{C}$. However, the properties of a homotopy initial functor are much
stronger than this. As suggested by the terminology, restriction along
$\varphi$ preserves arbitrary homotopy limits, as explained in (e) below.
(Alternatively, restriction along
$\varphi\colon\mathbf{D}^{\mathrm{op}}\rightarrow\mathbf{C}^{\mathrm{op}}$
preserves homotopy colimits.) Let us list several useful properties of such
functors, most of which can be seen as special instances of the last example
(e).

(a) If $X$ is a presheaf of sets on $\mathbf{C}$, with its restriction
$\varphi^{*}X$ a presheaf on $\mathbf{D}$, then the induced functor

$\mathbf{D}/\varphi^{*}X\rightarrow\mathbf{C}/X$

between the respective categories of elements is again homotopy initial (as is
easily verified), hence induces a homotopy equivalence

$B(\mathbf{D}/\varphi^{*}X)\xrightarrow{\simeq}B(\mathbf{C}/X).$

Note that this map is a pullback of the map $B\mathbf{D}\rightarrow
B\mathbf{C}$, but that the projection of the codomain to $B\mathbf{C}$ is
rarely a fibration.

(b) The same applies more generally for a fibered category
$p\colon\mathbf{E}\rightarrow\mathbf{C}$ and its pullback
$\varphi^{*}\mathbf{E}\rightarrow\mathbf{D}$. To prove this, one shows that
for $e$ an object of $\mathbf{E}$, the classifying space of
$\varphi^{*}\mathbf{E}/e$ is weakly equivalent to that of $\mathbf{D}/p(e)$ by
another application of Quillen’s Theorem A. (The directions are important
here; the result would not generally hold for opfibered categories.)

(c) Generalizing (a), one could take $X_{\bullet}$ to be a simplicial presheaf
on $\mathbf{C}$. Then

$\mathbf{D}/\varphi^{*}X_{\bullet}\rightarrow\mathbf{C}/X_{\bullet}$

is a map of bisimplicial sets that is a weak equivalence in each fixed
simplicial degree (of $X_{\bullet}$), hence induces a weak equivalence on
diagonals.

(d) If $A$ is a presheaf of abelian groups on $\mathbf{C}$, then $\varphi$
induces an isomorphism $H_{*}(\mathbf{C};A)\rightarrow
H_{*}(\mathbf{D};\varphi^{*}A)$ in homology. (Here homology is defined as the
collection of left derived functors of the colimit functor from abelian
presheaves on $\mathbf{C}$ to the category of abelian groups.) For this result
it is not necessary to require that $A$ is locally constant, as for ordinary
weak equivalences. Indeed, if $P_{\bullet}\rightarrow A$ is a resolution of
$A$ where $P_{\bullet}=\mathbb{Z}[X_{\bullet}]$ is free on a simplicial
presheaf $X_{\bullet}$, then the presheaf
$\varphi^{*}(P_{\bullet})=\mathbb{Z}[\varphi^{*}X_{\bullet}]$ is a similar
resolution of $X_{\bullet}$. The result then follows because for each
simplicial degree $p$, the map $H_{*}(\mathbf{C};\mathbb{Z}[X_{p}])\rightarrow
H_{*}(\mathbf{D};\mathbb{Z}[\varphi^{*}X_{p}])$ can be identified with

$H_{*}(\mathbf{C}/X_{p};\mathbb{Z}\Bigr{)}\xrightarrow{\simeq}H_{*}(\mathbf{D}/\varphi^{*}(X_{p});\mathbb{Z}),$

which is an isomorphism by case (a).

(e) Consider the category $\mathcal{E}^{\mathbf{C}^{\mathrm{op}}}$ of
presheaves with values in a simplicial model category $\mathcal{E}$, and
similarly $\mathcal{E}^{\mathbf{D}^{\mathrm{op}}}$. These presheaf categories
can be equipped with the projective model structures and $\varphi$ induces a
Quillen adjunction

${\mathcal{E}^{\mathbf{D}^{\mathrm{op}}}}$${\mathcal{E}^{\mathbf{C}^{\mathrm{op}}}.}$$\scriptstyle{\varphi_{!}}$$\scriptstyle{\varphi^{*}}$

Constructing a similar adjunction for the trivial functor
$\mathbf{C}\rightarrow 1$ gives the Quillen adjunction

${\mathcal{E}^{\mathbf{C}^{\mathrm{op}}}}$${\mathcal{E}}$$\scriptstyle{\mathrm{colim}_{\mathbf{C}}}$$\scriptstyle{\mathrm{const}}$

and similarly for $\mathbf{D}$. If $\varphi$ is homotopy initial, then there
is a natural isomorphism of derived functors (cf. [6, Section 19.6])

$\mathbf{L}\mathrm{colim}_{\mathbf{D}}\circ\mathbf{R}\varphi^{*}\cong\mathbf{L}\mathrm{colim}_{\mathbf{C}}.$

In other words, the restriction functor $\mathbf{R}\varphi^{*}$ preserves
homotopy colimits.

The proof of this is rather straightforward from the usual Bousfield–Kan
formula for homotopy colimits, which states that for an $\mathcal{E}$-valued
presheaf $X$ on $\mathbf{C}$ taking values in cofibrant objects, its homotopy
colimit may be computed as a tensor product

$X(-)\otimes_{\mathbf{C}}N(\mathbf{C}/-).$

Similarly, one computes the homotopy colimit of $\varphi^{*}X$ as

$X(\varphi(-))\otimes_{\mathbf{D}}N(\mathbf{D}/-).$

The latter is isomorphic to

$X(-)\otimes_{\mathbf{C}}N(\mathbf{C}/\varphi(-)),$

and the natural comparison map $N(\mathbf{C}/\varphi(-))\rightarrow
N(\mathbf{C}/-)$ is a weak homotopy equivalence of simplicial diagrams on
$\mathbf{C}$ by the assumption that $\varphi$ is homotopy initial. This
implies the result.

Examples (a) and (d) can be seen as special cases of this preservation of
homotopy colimits: for example (a), this relies on the fact that the
classifying space of the category of elements $\mathbf{C}/X$ is a model for
the homotopy colimit of the functor

$\mathbf{C}^{\mathrm{op}}\rightarrow\mathbf{Sets}\xrightarrow{\mathrm{const}}\mathbf{sSets}.$

(f) There is an $\infty$-categorical version of (e) above, stating that
precomposition by $\varphi$ preserves colimits with values in a fixed
$\infty$-category $\mathcal{E}$ (cf. [9, Theorem 4.1.3.1]). Also, example (b)
can be seen as a special instance of the fact the pullback of a homotopy
initial functor along a cartesian fibration is again homotopy initial [9,
Proposition 4.1.2.15].

## References

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# Infinite order solutions of second order linear differential equations

Garima Pant and Manisha Saini  department of mathematics, university of delhi,
delhi-110007, india<EMAIL_ADDRESS>department of mathematics,
university of delhi, delhi-110007, india<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract.

This article deals with the second order linear differential equations with
entire coefficients. We prove some results involving conditions on
coefficients so that the order of growth of every non-trivial solution is
infinite.

###### Key words and phrases:

entire function, meromorphic function, order of growth, differential equation

###### 2010 Mathematics Subject Classification:

34M10, 30D35

The research work of the second author is supported by research fellowship
from University Grants Commission (UGC), New Delhi, India.

The second author is Senior Research Fellow (UGC, New Delhi, India).

##  1\. Introduction

A well known result of Herold for second order linear differential equation

$f^{\prime\prime}+A(z)f^{\prime}+B(z)f=0$ (1)

with entire coefficients $A(z)$ and $B(z)$, where $B(z)\not\equiv 0$ says
that, all solutions are entire functions. As we know that the solutions of the
differential equation are local in nature so when we say them entire functions
it means we are talking in terms of their analytic continuation to the whole
plane. The order of growth of these solutions has been a topic of interest for
a long time.

H. Wittich has shown that solutions of equation (1) have order of growth
finite if and only if the coefficients $A(z)$ and $B(z)$ are polynomials. It
is obvious that if $A(z)$ or $B(z)$ is a transcendental entire function then,
the existence of non-trivial solutions with infinite order of growth is
guaranteed. It is interesting to note here that if one of the independent
solution is of infinite order and other is of finite order then, almost all
solutions of equation (1) are of infinite order. This leads to the question of
finding conditions on coefficients so that all non-trivial solutions of
equation (1) are of infinite order. To achieve these results the Nevanlinna’s
value distribution theory is the basic and fundamental tool. In this article,
we are going to use extensively the terms of value distribution theory such as
characteristic function $T(r,f)$, maximum modulus $M(r,f)$, order of growth
$\rho(f)$, hyper-order of growth $\rho_{2}(f)$ and exponent of convergence
$\lambda(f)$ of zeros of a meromorphic function $f$. All this and more can be
found in the well known books “Meromorphic functions”, “Value distribution
theory” and “Nevanlinna theory and complex differential equations” by W.K.
Hayman, Lo Yang and Ilpo Laine, respectively. The standard references for the
above paragarphs are [12], [16], [6] and [18].

In our recent work [8], we have obtained some conditions on the coefficients
of the equation (1) so that all non-trivial solutions of the equation have
order of growth infinite. These conditions are $\rho(A)\neq\rho(B)$ or $B(z)$
has Fabry gaps, where entire function $A(z)$ possesses zero as a Borel
exceptional value and $B(z)$ is transcendental entire function. In our first
result, we have simply replaced the zero as a Borel exceptional value by a
finite complex number as Borel exceptional value.

######  Theorem A.

Let $A(z)$ be an entire function with a finite Borel exceptional value and
$B(z)$ be a transcendental entire function satisfying any of the following
conditions

  1. (1)

$\rho(B)\neq\rho(A)$

  2. (2)

$B(z)$ has Fabry gaps.

Then, every non-trivial solution $f$ of equation (1) satisfies

$\rho(f)=\infty.$

Recall that an entire function
$f(z)=\sum_{n=0}^{\infty}a_{\lambda_{n}}z^{\lambda_{n}}$ has Fabry gaps if
$\lim_{n\to\infty}n/\lambda_{n}=0.$ One should note that the order of growth
of such function is non-zero [7].

We illustrate the theorem given above by some examples.

######  Example 1.

The differential equation

$f^{\prime\prime}+(e^{z^{2}}-1)f^{\prime}+e^{z}f=0$

has all non-trivial solutions of infinite order of growth.

######  Example 2.

All non-trivial solutions of the differential equation
$f^{\prime\prime}+e^{z}f^{\prime}+B(z)f=0$, where $B(z)$ is an entire function
with Fabry gaps, are of infinite order of growth.

Here, we give an example which shows that that the hypothesis of Theorem [A]
are necessary.

######  Example 3.

The differential equation

$f^{\prime\prime}+(e^{z}+1)f^{\prime}+e^{z}f=0$

possesses a solution $f(z)=e^{-z}$ which is of finite order of growth.

We know that there is a measure of order of growth associated with a function
of infinite order called its hyper-order of growth. The next main result of
this article gives the hyper-order of growth of the solutions of equation (1).

######  Theorem B.

Suppose that the coefficients $A(z)$ and $B(z)$ satisfy the hypothesis of
Theorem [A]. Then, all non-trivial solutions $f$ of equation (1) satisfy

$\rho_{2}(f)=\max\\{\rho(A),\rho(B)\\}$

where $\max\\{\rho(A),\rho(B)\\}$ is a finite quantity.

Now, we state a consequence of Theorem [A] and Theorem [B].

######  Corollary 1.

Suppose that $A(z)$ is an entire function with a finite Picard exceptional
value and $B(z)$ is an entire function satisfying the hypothesis of Theorem
[A]. Then, every non-trivial solution $f$ of equation (1) satisfies

$\rho(f)=\infty.$

Also,

$\rho_{2}(f)=\max\\{\rho(A),\rho(B)\\}$

when $\max\\{\rho(A),\rho(B)\\}$ is a finite quantity.

In our third result, we have relaxed the hypothesis of Theorem [A] and added
one additional factor to achieve the same conclusion as has been achieved in
Theorem [A]. We have also achieved the conclusion of Theorem [B] in terms of
hyper-order of growth as well.

######  Theorem C.

Suppose that $A(z)$ is an entire function with a finite Borel exceptional
value and $B(z)$ is a transcendental entire function such that
$\mu(B)\neq\rho(A)$. Then, all non-trivial solutions $f$ of equation (1)
satisfy

$\rho(f)=\infty.$

Also,

$\rho_{2}(f)=\max\\{\rho(A),\mu(B)\\}$

when $\max\\{\rho(A),\rho(B)\\}$ is a finite quantity.

Here, we would like to mention that the above result is a generalisation of a
theorem in our work [9]. The following result also holds true.

######  Corollary 2.

Let $A(z)$ be an entire function with finite Picard exceptional value and
$B(z)$ satisfies the hypothesis of Theorem [C]. Then, the conclusion of
Theorem [C] is also true.

The final result of this article is an extension of the work of Kwon and Kim
[11]. They have proved that if $A(z)$ has a finite deficient value and $B(z)$
is an entire function with $\rho(B)<1/2$, then all solutions of equation (1)
are of infinite order. We have removed the restriction on the order of $B(z)$
given by them.

######  Theorem D.

Suppose that $A(z)$ is transcendental entire function with a finite deficient
value and $B(z)$ is a transcendental entire function satisfying any of the
following conditions

  1. (1)

$B(z)$ has Fabry gaps

  2. (2)

$T(r,B)\sim\log{M(r,B)},r\to\infty$, outside a set of finite logarithmic
measure.

Then,

$\rho(f)=\infty$

where $f$ is a non-trivial solution of equation (1).

One can recall that an entire function
$f(z)=\sum_{n=0}^{\infty}a_{\lambda_{n}}z^{\lambda_{n}}$ has Fej$\acute{e}$r
gaps if

$\sum_{n}\lambda_{n}^{-1}<\infty.$

In [14], T. Murai showed that an entire function with Fej$\acute{e}$r gaps
satisfies

$T(r,f)\sim\log{M(r,f)}$

as $r\to\infty,$ outside a set of finite logarithmic measure. Therefore, the
following corollary is an easy consequence of Theorem [D].

######  Corollary 3.

Suppose that the coefficient $A(z)$ satisfies the hypothesis of Theorem [D]
and $B(z)$ is an entire function with Fej$\acute{e}$r gaps. Then, all non-
trivial solutions of equation (1) are of infinite order.

##  2\. Preliminary Results

To make this article self contained we list some results which will help in
reading the coming sections. For the statement of these results, we need the
notion of linear measure, logarithmic measure, upper logarithmic density and
lower logarithmic density. For a set $F\subset(0,\infty)$, we recall these
notions as

$m(F)=\int_{F}\,dt\qquad\qquad m_{l}(F)=\int_{F}\frac{1}{t}\,dt$
$\overline{\log dens}(F)=\limsup_{r\to\infty}\frac{m_{l}(F\cap[1,r))}{\log r}$
$\underline{\log dens}(F)=\limsup_{r\to\infty}\frac{m_{l}(F\cap[1,r))}{\log
r},$

respectively. It is obvious from the definition that $0\leq\underline{\log
dens}(F)\leq\overline{\log dens}(F)\leq 1$.

The following lemma is an important result and it has been used extensively
for proving the results of differential equations in complex domain.

######  Lemma 1.

[4] Let $f(z)$ be a transcendental meromorphic function and let
$\Gamma=\\{(k_{1},j_{1}),(k_{2},j_{2}),\ldots,(k_{m},j_{m})\\}$ denote finite
set of distinct pairs of integers that satisfy $k_{i}>j_{i}\geq 0$ for
$i=1,2,\ldots,m$. Let $\alpha>1$ and $\epsilon>0$ be given real constants.
Then,

  1. (1)

there exists a set $E_{1}\subset[0,2\pi)$ that has linear measure zero, such
that if $\psi_{0}\in[0,2\pi)\setminus E_{1},$ then there is a constant
$R_{0}=R_{0}(\psi_{0})>0$ so that for all $z$ satisfying $\arg z=\psi_{0}$ and
$|z|\geq R_{0}$ and for all $(k,j)\in\Gamma$, we have

$\left|\frac{f^{(k)}(z)}{f^{(j)}(z)}\right|\leq c\left(\frac{T(\alpha
r,f)}{r}\log^{\alpha}{r}\log{T(\alpha r,f)}\right)^{(k-j)}.$ (2)

If $f(z)$ is of finite order, then $f(z)$ satisfies

$\left|\frac{f^{(k)}(z)}{f^{(j)}(z)}\right|\leq|z|^{(k-j)(\rho(f)-1+\epsilon)}$
(3)

for all $z$ satisfying $\arg z=\psi_{0}$ and $|z|\geq R_{0}$ and for all
$(k,j)\in\Gamma$.

  2. (2)

there exists a set $E\subset(1,\infty)$ with $m_{l}(E)$ is finite and there
exists a constant $c>0$ that depends only on $\alpha$ and $\Gamma$ such that,
for all $z$ satisfying $|z|=r\notin E\cup[0,1]$ and for all $(k,j)\in\Gamma$
we have

$\left|\frac{f^{(k)}(z)}{f^{(j)}(z)}\right|\leq c\left(\frac{T(\alpha
r,f)}{r}\log^{\alpha}{r}\log{T(\alpha r,f)}\right)^{(k-j)}.$ (4)

If $f(z)$ is of finite order, then $f(z)$ satisfies:

$\left|\frac{f^{(k)}(z)}{f^{(j)}(z)}\right|\leq|z|^{(k-j)(\rho(f)-1+\epsilon)}$
(5)

for all $z$ satisfying $|z|\notin E\cup[0,1]$ and $|z|\geq R_{0}$ and for all
$(k,j)\in\Gamma$.

For the statement of next lemma we need to recall the following.

######  Definition 1.

[1, 8] For a non-constant polynomial $P(z)=a_{n}z^{n}+\ldots+a_{0}$, the term
$\delta(P,\theta)$ is defined as:
$\delta(P,\theta)=\operatorname{Re}(a_{n}e^{\iota n\theta})$, where
$\operatorname{Re}(z)$ denotes the real part of $z$. A ray
$\gamma=re^{\iota\theta}$ is called a critical ray of $e^{P(z)}$ when
$\delta(P,\theta)=0$.

The rays $\arg{z}=\theta$ such that $\delta(P,\theta)=0$ divides the complex
plane into $2n$ sectors of equal length $\pi/n$. Also, $\delta(P,\theta)>0$
and $\delta(P,\theta)<0$ in the alternative sectors. Suppose that
$0\leq\phi_{1}<\theta_{1}<\phi_{2}<\theta_{2}<\ldots<\phi_{n}<\theta_{n}<\phi_{n+1}=\phi_{1}+2\pi$
are $2n$ critical rays of $e^{P(z)}$ satisfying $\delta(P,\theta)>0$ for
$\phi_{i}<\theta<\theta_{i}$ and $\delta(P,\theta)<0$ for
$\theta_{i}<\theta<\phi_{i+1}$ where $i=1,2,3,\ldots,n$. For $\alpha$ and
$\beta$, we fix the notation as follows:

$\Omega(\alpha,\beta)=\\{z:\alpha\leq\arg{z}\leq\beta\\};0\leq\alpha<\beta\leq
2\pi.$

The following lemma established an estimate for a transcendental entire
function of integral order of growth.

######  Lemma 2.

[1] Let $A(z)=v(z)e^{P(z)}$ be an entire function, where $P(z)$ is a
polynomial of degree $n$ and $v(z)$ be an entire function with order less than
$n$. Then, for every $\epsilon>0$, there exists $E\subset[0,2\pi)$ of linear
measure zero such that

  1. (i)

for $\theta\in[0,2\pi)\setminus E$ such that $\delta(P,\theta)>0$, there
exists $R>1$ such that

$\exp\\{(1-\epsilon)\delta(P,\theta)r^{n}\\}\leq|A(re^{\iota\theta})|\leq\exp\\{(1+\epsilon)\delta(P,\theta)r^{n}\\}$
(6)

for $r>R.$

  2. (ii)

for $\theta\in[0,2\pi)\setminus E$ such that $\delta(P,\theta)<0$, there
exists $R>1$ such that

$\exp\\{(1+\epsilon)\delta(P,\theta)r^{n}\\}\leq|A(re^{\iota\theta})|\leq\exp\\{(1-\epsilon)\delta(P,\theta)r^{n}\\}$
(7)

for $r>R.$

Next lemma is from [3] and give estimate for an entire function of order less
than one.

######  Lemma 3.

Let $g(z)$ be an entire function of order $\rho$, where $0<\rho<\frac{1}{2}$
and let $\epsilon>0$ be a given constant. Then, there exists a set
$F\subset[0,\infty)$ that has upper logarithmic density at least $1-2\rho$
such that $|g(z)|>\exp(|z|^{\rho-\epsilon})$ for all $z$ satisfying $|z|\in
F.$

The following result is an easy consequence of
Phragm$\acute{\bf{e}}$n-Lindel$\ddot{\bf{o}}$f theorem.

######  Lemma 4.

[15] Suppose that $B(z)$ is an entire function with
$\rho(B)\in\big{[}1/2,\infty)$. Then there exists a sector
$\Omega(\alpha,\beta),$ $\beta-\alpha\geq\pi/\rho(B)$, such that

$\limsup_{r\rightarrow\infty}\frac{\log\log|B(re^{\iota\theta})|}{\log
r}\geq\rho(B)$

for all $\theta\in\Omega(\alpha,\beta)$, where $0\leq\alpha<\beta\leq 2\pi.$

Next lemma give property of an entire function with Fabry gaps and can be
found in [13], [17].

######  Lemma 5.

Let $g(z)=\sum_{n=0}^{\infty}a_{\lambda_{n}}z^{\lambda_{n}}$ be an entire
function of finite order with Fabry gaps, and $h(z)$ be an entire function
with $\rho(h)=\sigma\in(0,\infty)$. Then for any given
$\epsilon\in(0,\sigma)$, there exists a set $F\subset(1,+\infty)$ satisfying
$\overline{\log dens}(F)\geq\xi$, where $\xi\in(0,1)$ is a constant such that
for all $|z|=r\in F$, one has

$\log M(r,h)>r^{\sigma-\epsilon},\quad\log m(r,g)>(1-\xi)\log M(r,g),$

where $M(r,h)=\max\\{|h(z)|:|z|=r\\}$, $m(r,g)=\min\\{|g(z)|:|z|=r\\}$ and
$M(r,g)=\max\\{|g(z)|:|z|=r\\}$.

The following remark follows from the above lemma.

######  Remark 1.

Suppsoe that $g(z)=\sum_{n=0}^{\infty}a_{\lambda_{n}}z^{\lambda_{n}}$ be an
entire function of order $\sigma\in(0,\infty)$ with Fabry gaps then for any
given $\epsilon>0,\quad(0<2\epsilon<\sigma)$, there exists a set
$F\subset(1,+\infty)$ satisfying $\overline{\log dens}(F)\geq\xi$, where
$\xi\in(0,1)$ is a constant such that for all $|z|=r\in F$ , one has

$|g(z)|>M(r,g)^{(1-\xi)}>\exp{\left((1-\xi)r^{\sigma-\epsilon}\right)}>\exp{\left(r^{\sigma-2\epsilon}\right)}.$

The next lemma gave the estimate for a meromorphic function.

######  Lemma 6.

[11] Suppose that $g$ is a meromorphic function of order $\rho\in[0,\infty)$.
For given $\zeta>0$ and $0<l<1/2$, there exists a constant $K(\rho,\zeta)$ and
a set $F_{\zeta}\subset[0,\infty)$ of lower logarithmic density greater than
$1-\zeta$ such that

$r\int_{J}\bigg{|}\frac{g^{\prime}(re^{\iota\theta})}{g(re^{\iota\theta})}\,d\theta\bigg{|}<K(\rho,\zeta)\bigg{(}l\log{\frac{1}{l}}\bigg{)}T(r,g)$

for all $r\in F_{\zeta}$ and for every interval $J\subset[0,2\pi)$ of length
$l$.

The next lemma provided an upper bound for the hyper-order of growth of
solutions $f$ of equation (1).

######  Lemma 7.

[19] Suppose that $A(z)$ and $B(z)$ are entire functions of finite order.
Then,

$\rho_{2}(f)\leq\max\\{\rho(A),\rho(B)\\}$

for all solutions $f$ of equation (1).

The following lemma involves the central index of a transcendental entire
function.

######  Lemma 8.

[12] Let $f$ be a transcendental entire function, $0<\delta<1/4$ and $z$ be
such that $|z|=r$ and $|f(z)|>M(r,f)\nu(r,f)^{-\frac{1}{4}+\delta}$ holds.
Then there exists a set $F\subset(0,\infty)$ with $m_{l}(F)<\infty$ such that

$f^{(m)}(z)=\left(\frac{\nu(r,f)}{z}\right)^{m}(1+o(1))f(z)$ (8)

for all $m\geq 0$ and all $r\notin F$.

##  3\. Results

This section includes those results which we have framed with proofs for
getting foothold in the last section.

######  Lemma 9.

Suppose that $f$ is an entire function such that $T(r,f)\sim\log{M(r,f)}$ as
$r\to\infty,$ outside a set of finite logarithmic measure. For $0<c<1$, let

$E(r)=\\{\theta\in[0,2\pi):\log|f(re^{\iota\theta})|\leq(1-c)\log{M(r,f)}\\}$

then, $m(E(r))\to 0$ as $r\to\infty$, outside a set of finite logarithmic
measure.

###### Proof.

This result is already in arxiv [2001.10729]. Proof can be seen there. ∎

We present a result which follows easily from Lemma [8].

######  Lemma 10.

Suppose $f$ is a transcendental entire funtion then, there exists a set
$F\subset(0,\infty)$ with finite logarithmic measure such that for all $z$
satisfying $|z|=r\notin F$ and $|f(z)|=M(r,f)$ we have

$\bigg{|}\frac{f(z)}{f^{(m)}(z)}\bigg{|}\leq 2r^{m}$ (9)

for all $m\in\mathbb{N}.$

###### Proof.

Let $z$ be a point on the circle $|z|=r$ such that $|f(z)|=M(r,f)$ then, Lemma
[8] implies that, there exists $F\subset(0,\infty)$ with finite logarithmic
measure such that

$f^{(m)}(z)=\left(\frac{\nu(r,f)}{z}\right)^{m}(1+o(1))f(z),\quad
m\in\mathbb{N}$

for all $r\notin F$. We observe that $\nu(r,f)\geq 1$ therefore, equation (8)
implies that

$\bigg{|}\frac{f(z)}{f^{(m)}(z)}\bigg{|}\leq\frac{|z|^{m}}{\nu(r,f)}(1+o(1))\leq
2r^{m}$

for all $m\in\mathbb{N}$ and $r\notin F$. ∎

The next lemma shows that the conclusion of Lemma [10] holds true in a
neighbourhood of the argument of $z$.

######  Lemma 11.

Let $f$ be a transcendental entire function and $z_{r}=re^{\iota\theta_{r}}$
be a point such that $|f(z_{r})|=M(r,f)$. Then, for sufficiently large
$r\notin F$ where $m_{l}(F)<\infty$, there exists a constant $\delta_{r}>0$
such that for all $z$ satisfying $|z|=r$ and
$\arg{z}=\theta\in[\theta_{r}-\delta_{r},\theta_{r}+\delta_{r}]$, one has

$\bigg{|}\frac{f(z)}{f^{(m)}(z)}\bigg{|}\leq 2r^{m}$

for all $m\in\mathbb{N}.$

###### Proof.

The point $z_{r}=re^{\iota\theta_{r}}$ satisfies $|f(z_{r})|=M(r,f)$. The
function $|f(z)|$ is continuous on $|z|=r$ thus, there exists $\delta_{r}$
such that for all $z$ satisfying $|z|=r$ and
$\arg{z}=\theta\in[\theta_{r}-\delta_{r},\theta_{r}+\delta_{r}]$, we have

$||f(z)|-|f(z_{r})||<\frac{|f(z_{r})|}{2}.$

This again implies

$|f(z)|>\frac{|f(z_{r})|}{2}=\frac{M(r,f)}{2}>M(r,f)\nu(r,f)^{-1/4+\delta}$

for some $0<\delta<1/4$. Therefore, Lemma [8] yields that

$f^{(m)}(z)=\left(\frac{\nu(r,f)}{z}\right)^{m}(1+o(1))f(z),\quad
m\in\mathbb{N}$

for all $|z|=r\notin F$ and
$\arg{z}=\theta\in[\theta_{r}-\delta_{r},\theta_{r}+\delta_{r}]$. Now, the
conclusion follows from Lemma [10]. ∎

##  4\. Proofs of Theorems

######  Proof of Theorem A.

Let $a\in\mathbb{C}$ be the Borel exceptional value of the function $A(z)$.
Therefore, the function $A(z)-a$ is an entire function which has zero as its
Borel exceptional value. By employing the Weierstrass factorisation theorem,
we can write

$A(z)-a=v(z)e^{P(z)}$ (10)

where $P(z)$ is a non-constant polynomial of degree $n$ and $v(z)(\not\equiv
0)$ is an entire function such that $\rho(v)<n$. Lemma [2] implies that for
$\epsilon>0$ there exists $E\subset[0,2\pi)$ with $m(E)=0$ such that for
$\theta\in[0,2\pi)\setminus E$ with $\delta(P,\theta)<0$ there exists
$R_{0}>1$ such that

$|A(z)-a|\leq\exp\\{(1-\epsilon)\delta(P,\theta)r^{n}\\}$ (11)

for all $z=re^{\iota\theta}$ and $r>R_{0}$. Suppose that $f$ is a finite order
solution of the equation (1). Therefore, by Lemma 1, there exists a set
$E_{1}\subset[0,2\pi)$ with $m(E_{1})=0$, such that if
$\theta\in[0,2\pi)\setminus E_{1}$, then there is a constant $R=R(\theta)$ so
that for $z$ satisfying $\arg{z}=\theta$ and $|z|>R$, we have

$\big{|}\frac{f^{(k)}(z)}{f(z)}\big{|}\leq|z|^{2\rho(f)},k=1,2.$ (12)

  1. (1)

Suppose that $\rho(A)<\rho(B)$ then, the conclusion holds using the Theorem
[2] in [5]. Therefore, we need to consider $\rho(B)<\rho(A)$. The following
three cases are there to be discussed:

    1. (a)

when $0<\rho(B)<1/2$ then, by the Lemma 3, for $0<\alpha<\rho(B)$ there exists
a set $F\subset(0,\infty)$ with $\overline{\log dens}(F)>1-\rho(B)$ such that

$|B(re^{\iota\theta})|>\exp\\{r^{\alpha}\\}$ (13)

for all $r\in F$. From the equation (1), (11), (12) and (13) for all $z$
satisfying $\theta\in[0,2\pi)\setminus\big{(}E_{1}\cup E\big{)}$ with
$\delta(P,\theta)<0$ and $|z|=r\in F$, $r>R_{0}$ we have

$\displaystyle\exp\\{r^{\alpha}\\}$ $\displaystyle<|B(re^{\iota\theta})|$
$\displaystyle\leq\big{|}\frac{f^{\prime\prime}(re^{\iota\theta})}{f(re^{\iota\theta})}\big{|}+|A(re^{\iota\theta})|\big{|}\frac{f^{\prime}(re^{\iota\theta})}{f(re^{\iota\theta})}\big{|}$
$\displaystyle\leq r^{2\rho(f)}\big{(}1+|A(re^{\iota\theta})-a|+|a|\big{)}$
$\displaystyle\leq r^{2\rho(f)}\big{(}1+o(1)+|a|\big{)}$

which implies a contradiction for arbitrary large $r$.

    2. (b)

If $\rho(B)\in[\frac{1}{2},\infty)$ then, using Lemma [4], there exists a
sector $\Omega(\alpha,\beta);0\leq\alpha<\beta\leq 2\pi$ with
$\beta-\alpha\geq\frac{\pi}{\rho(B)}$ such that

$\limsup_{r\to\infty}\frac{\log\log|B(re^{\iota\theta})|}{\log{r}}=\rho(B)$
(14)

for all $\theta\in[\alpha,\beta]$. We have that $\rho(B)<\rho(A)$, therefore
there exists $\theta_{0}\in[\alpha,\beta]\setminus\big{(}E\cup E_{1}\big{)}$
such that the equation (11) holds true for $\theta_{0}$. We get a
contradiction using equations (1), (11), (12) and (14).

    3. (c)

When $\rho(B)=0$ then, a result of [2] implies that

$\limsup_{r\to\infty}\frac{\log|B(re^{\iota\theta})|}{\log{r}}=\infty$ (15)

for all $\theta\in[0,2\pi)$. This will implies that

$r^{M}<|B(re^{\iota\theta})|$ (16)

for all $r>R(M)$ and $\theta\in[0,2\pi)$, where $M$ is an arbitrary large
constant. Therefore, for all sufficiently large $r$ and
$\theta_{0}\in[0,2\pi)\setminus\big{(}E\cup E_{1}\big{)}$ with
$\delta(P,\theta_{0})<0$, using equations (1), (11), (12) and (16), we get a
contradiction.

  2. (2)

Suppose that $B(z)$ is a function with Fabry gaps. Then, Remark [1] for
$0<\alpha<\rho(B)$, there exists a set $F\subset(0,\infty)$ with
$\overline{\log dens}(F)>0$ such that

$|B(re^{\iota\theta}|>\exp\\{r^{\alpha}\\}$ (17)

for all $r\in F$. Therefore, for sufficiently large $r\in F$ and
$\theta_{0}\in[0,2\pi)\setminus\big{(}E\cup E_{1})$ with
$\delta(P,\theta_{0})<0$, by the equations (1), (11), (12) and (17), we get a
contradiction.

∎

######  Proof of Theorem B.

  1. (1)

We know that all solutions $f(\not\equiv 0)$ of equation (1) are of infinite
order, when $\rho(B)\neq\rho(A)$ by Theorem [A]. Using Lemma [1], for
$\epsilon>0$, there exists a set $E\subset[1,\infty)$ that has finite
logarithmic measure such that for all $z$ satisfying $|z|=r\notin E\cup[0,1]$
we have

$\left|\frac{f^{(k)}(z)}{f^{(j)}(z)}\right|\leq
c\left[T(2r,f)\right]^{2(k-j)}$ (18)

where $c>0$ is a constant.

If $\rho(A)<\rho(B)$ then from [10, Theorem 1] and Lemma [7] we get that
$\rho_{2}(f)=\max\\{\rho(A),\rho(B)\\}$.

If $\rho(B)<\rho(A)=n,n\in\mathbb{N},$ one can choose $\beta$ such that
$\rho(B)<\beta<\rho(A)$. Suppose that $a\in\mathbb{C}$ is a Borel exceptional
value of $A(z)$. Therefore, $A(z)-a=v(z)e^{P(z)}$, where $P(z)$ is a non-
constant polynomial of degree $n$ and $v(z)$ is an entire function with
$\rho(v)<n$.

From Lemma [10], we can choose a sequence $r_{m}\notin E\cup[0,1]\cup F$ such
that $r_{m}\to_{m\to\infty}\infty$ and let
$|f(r_{m}e^{\iota\theta_{m}})|=M(r_{m},f)$. Using Lemma [10], we obtain

$\bigg{|}\frac{f(r_{m}e^{\iota\theta_{m}})}{f^{\prime}(r_{m}e^{\iota\theta_{m}})}\bigg{|}\leq
2r_{m}.$ (19)

Suppose that there exists a subsequence $(\theta_{m})$ such that
$\lim_{m\to\infty}\theta_{m}=\theta_{0}.$ We have the following cases to
discuss.

    1. (i)

If $\delta(P,\theta_{0})>0$ then, the continuity of $\delta(P,\theta)$ implies
that

$\frac{1}{2}\delta(P,\theta_{0})<\delta(P,\theta_{m})<\frac{3}{2}\delta(P,\theta_{0})$

for all sufficiently large $m\in\mathbb{N}$. The part (i) of Lemma [2] gives

$\displaystyle\exp{\left(\frac{1}{2}(1-\epsilon)\delta(P,\theta_{0})r^{n}_{m}\right)}$
$\displaystyle\leq|A(r_{m}e^{\iota\theta_{m}})-a|$ (20)
$\displaystyle\leq\exp{\left(\frac{3}{2}(1+\epsilon)\delta(P,\theta_{0})r^{n}_{m}\right)}$

for all sufficiently large $m\in\mathbb{N}$. The equations (1), (19), (20) and
(18), for $z_{m}=r_{m}e^{\iota\theta_{m}}$ implies that

$\displaystyle\qquad\exp{\left(\frac{1}{2}(1-\epsilon)\delta(P,\theta_{0})r^{n}_{m}\right)}$
$\displaystyle\leq|A(r_{m}e^{\iota\theta_{m}})-a|$
$\displaystyle\leq\left|\frac{f^{\prime\prime}(r_{m}e^{\iota\theta_{m}})}{f^{\prime}(r_{m}e^{\iota\theta_{m})}}\right|$
$\displaystyle+|B(r_{m}e^{\iota\theta_{m}})|\left|\frac{f(r_{m}e^{\iota\theta_{m}})}{f^{\prime}(r_{m}e^{\iota\theta_{m}})}\right|+|a|$
$\displaystyle\leq
c\left[T(2r_{m},f)\right]^{2}+\exp{\left(r_{m}^{\beta}\right)}2r_{m}+|a|.$

As $\beta<n,$ this implies that

$\limsup_{m\rightarrow\infty}\frac{\log^{+}\log^{+}T(r_{m},f)}{\log{r_{m}}}\geq\rho(A).$
(21)

Using Lemma [7] and equation (21) we have

$\rho_{2}(f)=\max\\{\rho(A),\rho(B)\\}.$

    2. (ii)

If $\delta(P,\theta_{0})<0$ then, the continuity of $\delta(P,\theta)$ implies
that

$\frac{3}{2}\delta(P,\theta_{0})<\delta(P,\theta_{m})<\frac{1}{2}\delta(P,\theta_{0})$

for sufficiently large $m\in\mathbb{N}$. Using part (ii) of Lemma [2] we
obtain

$\displaystyle\exp{\left(\frac{3}{2}(1+\epsilon)\delta(P,\theta_{0})r^{n}_{m}\right)}$
$\displaystyle\leq|A(r_{m}e^{\iota\theta_{m}})-a|$ (22)
$\displaystyle\leq\exp{\left(\frac{1}{2}(1-\epsilon)\delta(P,\theta_{0})r^{n}_{m}\right)}$

for all sufficiently large $m\in\mathbb{N}$. The equations (1), (18) (19) and
(22), for $z_{m}=r_{m}e^{\iota\theta_{m}}$ implies that

$\displaystyle\qquad\exp{\left(\frac{3}{2}(1+\epsilon)\delta(P,\theta_{0})r^{n}_{m}\right)}$
$\displaystyle\leq|A(z_{m})-a|$
$\displaystyle\leq\left|\frac{f^{\prime\prime}(r_{m}e^{\iota\theta_{m}})}{f^{\prime}(r_{m}e^{\iota\theta_{m})}}\right|$
$\displaystyle+|B(r_{m}e^{\iota\theta_{m}})|\left|\frac{f(r_{m}e^{\iota\theta_{m}})}{f^{\prime}(r_{m}e^{\iota\theta_{m}})}\right|+|a|$
$\displaystyle\leq
c\left[T(2r_{m},f)\right]^{2}+\exp{\left(r_{m}^{\beta}\right)}2r_{m}+|a|.$

Therefore, as earlier we have $\rho_{2}(f)=\max\\{\rho(A),\rho(B)\\}$.

    3. (iii)

If $\delta(P,\theta_{0})=0$. For all $m\in\mathbb{N}$, we use Lemma [11] to
have an interval $(\theta_{m}-l_{0},\theta_{m}+l_{0}),0<l_{0}<1/2$ such that,
for all $\theta\in(\theta_{m}-l_{0},\theta_{m}+l_{0})$ we have

$\frac{|f(r_{m}e^{\iota\theta})|}{|f^{\prime}(r_{m}e^{\iota\theta})|}\leq
2r_{m}.$ (23)

Again choose $\theta^{*}_{m}$ such that
$\theta^{*}_{m}\in(\theta_{m}+l_{0}/3,\theta_{m}+l_{0})$ and
$\theta^{*}_{m}\to\theta^{*}_{0}$ as $m\to\infty$. This implies
$\theta^{*}_{0}\in(\theta_{0}+l_{0}/3,\theta_{0}+l_{0})$.

Let us consider $\delta(P,\theta^{*}_{0})>0$ then, again by the continuity of
$\delta(P,\theta)$ we have

$\displaystyle\exp{\left(\frac{1}{2}(1-\epsilon)\delta(P,\theta_{0}^{*})r^{n}_{m}\right)}$
$\displaystyle\leq|A(r_{m}e^{\iota\theta^{*}_{m}})-a|$ (24)
$\displaystyle\leq\exp{\left(\frac{3}{2}(1+\epsilon)\delta(P,\theta_{0}^{*})r^{n}_{m}\right)}$

for sufficiently large $m$. Applying equations (1), (18), (23) and (24) , for
$z^{*}_{m}=r_{m}e^{\iota\theta^{*}_{m}}$, we get the desired result.

If $\delta(P,\theta^{*}_{0})<0$ then, as in the case (ii) we can show that the
conclusion holds true .

  2. (2)

It has been already proved that all non-trivial solutions $f(z)$ of equation
(1), with $A(z)$ and $B(z)$ satisfying the hypothesis of Theorem [A], are of
infinite order. Also, if $\rho(A)\neq\rho(B)$ then, using case (1) above,

$\rho_{2}(f)=\max\\{\rho(A),\rho(B)\\}.$

Now, let $\rho(A)=\rho(B)=n,n\in\mathbb{N}$. Using Remark [1], for
$\epsilon>0$, there exist $F\subset(1,\infty)$ satisfying $\overline{\log
dens}(F)>0$ such that for all $|z|=r\in F$ we have

$|B(z)|>\exp{\\{r^{n-\epsilon}\\}}.$ (25)

Now, we choose $\theta\in[0,2\pi)$ with $\delta(P,\theta)<0$ and a sequence
$(r_{m})\subset F\setminus\left(E\cup[0,1]\right)$ satisfying
$r_{m}\to\infty.$ Using equations (1), (7), (18) and (25) we obtain

$\displaystyle\qquad\quad\exp{\left(r_{m}^{n-\epsilon}\right)}$
$\displaystyle<|B(r_{m}e^{\iota\theta})|\leq\left|\frac{f^{\prime\prime}(r_{m}e^{\iota\theta})}{f(r_{m}e^{\iota\theta})}\right|+|A(r_{m}e^{\iota\theta})|\left|\frac{f^{\prime}(r_{m}e^{\iota\theta})}{f(r_{m}e^{\iota\theta})}\right|$
$\displaystyle\leq cT(2r_{m},f)^{4}$
$\displaystyle+(\exp{((1-\epsilon)\delta(P,\theta)r_{m}^{n})}+|a|)cT(2r_{m},f)^{2}$
$\displaystyle\leq cT(2r_{m},f)^{2}(1+o(1)).$

Thus, we conclude that

$\limsup_{m\rightarrow\infty}\frac{\log^{+}\log^{+}T(r_{m},f)}{\log r_{m}}\geq
n.$ (26)

Using Lemma [7] and equation (26) we get,

$\rho_{2}(f)=\max\\{\rho(A),\rho(B)\\}.$

∎

######  Proof of Theorem C.

Let $a\in\mathbb{C}$ is a finite Borel exceptional value then,
$A(z)-a=v(z)e^{P(z)}$ where $P(z)$ is a non-constant polynomial and $v(z)$ is
an entire function with $\rho(v)<\deg{P}.$ Now, we can apply the arguments of
Theorem [1] in [9] to get the conclusion of the theorem.

∎

######  Proof of Theorem D.

Suppose $a\in\mathbb{C}$ is the finite deficient value of the function $A(z)$.
Thus,

$\liminf_{r\to\infty}\frac{m\left(r,\frac{1}{A-a}\right)}{T(r,A)}=2d>0.$

This yields that, for all sufficiently large $r$, we have

$m\left(r,\frac{1}{A-a}\right)>dT(r,A).$

Thus, for all sufficiently large $r$, there exists a $\theta_{r}$ such that
for $z_{r}=re^{\iota\theta_{r}}$, one may have

$\log{|A(z_{r})-a|}\leq-dT(r,A).$ (27)

In Lemma [6], let $\zeta>0$ and choose $0<l<1/2$ so that
$K(\rho(A),\zeta)\big{(}l\log{\frac{1}{l}}\big{)}$ is sufficiently small and
choose $\phi>0$ with $|\theta_{r}-\phi|\leq l$ such that

$\displaystyle\log{|A(re^{\iota\theta})-a|}$
$\displaystyle=\log{|A(re^{\iota\theta_{r}})-a|}+\int_{\theta_{r}}^{\theta}\frac{d}{dt}\log{|A(re^{\iota
t})-a|}$ $\displaystyle\leq-
dT(r,A)+r\int_{\theta_{r}}^{\theta}\bigg{|}\frac{(A-a)^{\prime}(re^{\iota
t})}{(A-a)(re^{\iota t})}\bigg{|}\,dt$ $\displaystyle\leq-
dT(r,A)+K(\rho(A),\zeta)\bigg{(}l\log{\frac{1}{l}}\bigg{)}T(r,A)$
$\displaystyle\leq 0$

for all sufficiently large $r\in F_{\zeta}$ where $\underline{\log
dens}(F_{\zeta})\geq 1-\zeta$ and for all
$\theta\in[\theta_{r}-\phi,\theta_{r}+\phi]$. Hence, for all sufficiently
large $r\in F_{\zeta}$ and $\theta\in[\theta_{r}-\phi,\theta_{r}+\phi]$ we
have

$|A(re^{\iota\theta})|\leq|a|+1.$ (28)

  1. (1)

Let $B(z)$ be a transcendental entire function with Fabry gaps. Then, Remark
[1] implies that for $0<\alpha<\rho(B)$, there exists $F\subset(0,\infty)$
such that $\overline{\log dens}(F)>0$ and equation (17) holds true for all
$r\in F$. We choose $\zeta>0$ sufficiently small such that $\overline{\log
dens}(F)>\zeta$. We know that

$\chi_{F\cap F_{\zeta}}=\chi_{F}+\chi_{F_{\zeta}}-\chi_{F\cup F_{\zeta}}$

and $\overline{\log dens}(F\cup F_{\zeta})\leq 1$, therefore

$\displaystyle\overline{\log dens}(F\cap F_{\zeta})$
$\displaystyle\geq\overline{\log dens}(F)+\underline{\log
dens}(F_{\zeta})-\overline{\log dens}(F\cup F_{\zeta})$
$\displaystyle\geq\overline{\log dens}(F)+1-\zeta-1$ $\displaystyle>0.$

Hence, for all $r\in F\cap F_{\zeta}$ and
$\theta\in[\theta_{r}-\phi,\theta_{r}+\phi]\setminus E_{1}$, applying
equations (1), (12), (17) and (28) we get

$\displaystyle\exp\\{r^{\alpha}\\}$ $\displaystyle<|B(re^{\iota\theta})|$
$\displaystyle\leq\bigg{|}\frac{f^{\prime\prime}(re^{\iota\theta})}{f(re^{\iota\theta})}\bigg{|}+|A(re^{\iota\theta})|\bigg{|}\frac{f^{\prime}(re^{\iota\theta})}{f(re^{\iota\theta})}\bigg{|}$
$\displaystyle\leq r^{2\rho(f)}\big{(}1+|a|+1\big{)}.$

This is a contradiction for sufficiently large $r\in F\cap F_{\zeta}.$

  2. (2)

Suppose that $B(z)$ is a transcendental entire function satisfying

$T(r,B)\sim\log M(r,B))$

outside a set $H$ of finite logarithmic measure. Using Lemma [9], for a given
$0<c<1$, we get

$\log{|B(re^{\iota\theta})|}\geq(1-c)\log{M(r,B)}$ (29)

for sufficiently large $r\notin H$ and $\theta\notin J(r)$. Applying equations
(1), (12), (28) and (29), for all sufficiently large $r\in F_{\zeta}\setminus
H$ and $\theta\in[\theta_{r}-\phi,\theta_{r}+\phi]\setminus(E_{1}\cup J(r))$,
we get

$\displaystyle M(r,B)^{1-c}$ $\displaystyle\leq|B(re^{\iota\theta})|$
$\displaystyle\leq\bigg{|}\frac{f^{\prime\prime}(re^{\iota\theta})}{f(re^{\iota\theta})}\bigg{|}+|A(re^{\iota\theta})|\bigg{|}\frac{f^{\prime}(re^{\iota\theta})}{f(re^{\iota\theta})}\bigg{|}$
$\displaystyle\leq r^{2\rho(f)}\big{(}1+|a|+1\big{)}.$

This gives that $M(r,B)\leq r^{4\rho}(2+|a|)^{2}$ for sufficiently large $r\in
F_{\zeta}\setminus H$. This will be a contradiction, as $B(z)$ is a
transcendental entire function.

∎

Acknowledgement: We are thankful to our thesis advisor Sanjay Kumar for
valuable suggestions.

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