# Correlated noise enhances coherence and fidelity in coupled qubits

Eric R. Bittner  Department of Chemistry, University of Houston, Houston,
Texas 77204, United States.  Center for Nonlinear Studies, Los Alamos National
Laboratory, Los Alamos, NM 87545, United States<EMAIL_ADDRESS>Hao
Li  Department of Chemistry, University of Houston, Houston, Texas 77204,
United States.  S. A. Shah  Center for Nonlinear Studies, Los Alamos National
Laboratory, Los Alamos, NM 87545, United States  Theoretical Division, Los
Alamos National Laboratory, Los Alamos, NM 87545, United States  Carlos Silva
School of Chemistry and Biochemistry, Georgia Institute of Technology,
Atlanta, GA 30332, United States  School of Physics, Georgia Institute of
Technology, Atlanta, GA 30332, United States  School of Materials Science and
Engineering, Georgia Institute of Technology, Atlanta, GA 30332, United States
Andrei Piryatinski  Theoretical Division, Los Alamos National Laboratory, Los
Alamos, NM 87545, United States

###### Abstract

It is generally assumed that environmental noise arising from thermal
fluctuations is detrimental to preserving coherence and entanglement in a
quantum system. In the simplest sense, dephasing and decoherence are tied to
energy fluctuations driven by coupling between the system and the normal modes
of the bath. Here, we explore the role of noise correlation in an open-loop
model quantum communication system whereby the “sender” and the “receiver” are
subject to local environments with various degrees of correlation or
anticorrelation. We introduce correlation within the spectral density by
solving a multidimensional stochastic differential equations and introduce
these into the Redfield equations of motion for the system density matrix. We
find that correlation can enhance both the fidelity and purity of a maximally
entangled (Bell) state. Moreover, we show that, by comparing the evolution of
different initial Bell states, one can effectively probe the correlation
between two local environments. These observations may be useful in the design
of high-fidelity quantum gates and communication protocols.

##  I Introduction

Noise and environmental fluctuations are generally considered detrimental to
the preservation of coherence and entanglement in an open quantum system.
Correlations between individual quantum systems represent the basic resources
in quantum information and quantum computing, and one of the major
technological tasks is to protect and control these correlations and
entanglements. Entanglement expresses the non-separability of the quantum
state of a compound system. However, the coupling to environment leads to
dissipation and loss of the quantum correlations, often on time scales much
shorter than those needed for implementing quantum information tasks. Such
fluctuations can arise from nuclear and electronic motions of the surrounding
environment that induce a noisy driving field. From the Anderson-Kubo (AK)
model for spectral line-shapes [1, 2, 3, 4], the transition frequency obeys an
Ornstein-Uhlenbeck (OU) process such that
$\omega_{t}=\omega_{o}+\delta\omega_{t}$ and

$\displaystyle d\delta\omega_{t}=-\gamma\delta\omega_{t}dt+\sigma dW_{t}$ (1)

with $\langle\delta\omega(t)\rangle=0$ and

$\displaystyle\langle\delta\omega(t)\delta\omega(0)\rangle=\frac{\sigma^{2}}{2\gamma}e^{-\gamma|t|}$
(2)

where $W_{t}$ is the Wiener process, $\Delta^{2}=\frac{\sigma^{2}}{2\gamma}$
is the fluctuation amplitude, and ${1}/{\gamma}=\tau_{c}$ is the correlation
time for the noise. According to the AK model, $\Delta\tau_{c}\ll 1$
corresponds to the case of fast modulation, which results in a purely
Lorentzian spectral line shape and a pure dephasing time of
$T_{2}=(\Delta^{2}\tau_{c})^{-1}$. Similarly, when $\Delta\tau_{c}\gg 1$, we
are in the slow modulation regime and the spectral lineshape takes a purely
Gaussian form reflecting the inhomogeneities of the environment. We recently
extended this approach to account for non-stationary/non-equilibrium
environments. [5, 6]

Actively preventing decoherence from affecting quantum entanglement holds both
theoretical and practical significance in quantum information processing
technologies. Several recent studies suggest that decoherence can be
suppressed by carefully engineering the system-bath coupling. [7, 8, 9] For
example, Mouloudakis and Lambropoulos, extending previous work by Yang et
al.[10], studied the steady-state entanglement between two qubits that
interacted asymmetrically with a common non-Markovian environment. The study
found that, depending on the initial two-qubit state, the asymmetry in the
couplings between each qubit and the non-Markovian environment could lead to
enhanced entanglement in the steady state of the system.[11, 12]

However, it is possible, especially in a condensed phase environment, that
multiple modes of the environment can contribute to the frequency
fluctuations, and it is possible that these contributions can be correlated,
anti-correlated, or totally uncorrelated. To set the stage for our subsequent
analysis, let us consider what happens if we extend the AK model to account
for multiple noise contributions. Consider a single stochastic process,
$E_{t}$, described by a generalization of the Ito stochastic differential
equation (SDE)[13],

$\displaystyle dE_{t}=-\gamma E_{t}dt+B\cdot dW_{t},$ (3)

where $B$ is a vector of variances $B=\\{\sigma_{1},\sigma_{2}\\}$, and
$dW=\\{dW_{1},dW_{2}\\}$ are correlated Wiener processes with
$dW_{1}(t)dW_{2}(t^{\prime})=\delta(t-t^{\prime})\xi dt$ and
$dW_{i}(t)dW_{i}(t^{\prime})=\delta(t-t^{\prime})dt$, where $-1\leq\xi\leq 1$
is the correlation parameter between the two Wiener processes. We can rewrite
the SDE in Eq. 3 in terms of two uncorrelated processes by defining the
variances
$B^{\prime}=\\{\sigma_{1}+\sigma_{2}\xi,\sigma_{2}(1-\xi^{2})^{1/2}\\}$ such
that Eq. 3 becomes

$\displaystyle dE_{t}=-\gamma
E_{t}dt+(\sigma_{1}+\sigma_{2}\xi)dW_{1}+\sigma_{2}(1-\xi^{2})^{1/2}dW_{2}$
(4)

and $W_{1}$ and $W_{2}$ are now uncorrelated Wiener processes. If we work out
the covariance of $E_{t}$ one finds that

$\displaystyle{Cov}[E_{t},E_{s}]=\frac{e^{-\gamma(s+t)}}{2\gamma}(e^{2\gamma{\rm
min}(s,t)}-1)\sigma_{\mathrm{eff}}^{2}$ (5)

where we can define an effective covariance parameter

$\displaystyle\sigma_{\mathrm{eff}}^{2}=\sigma_{1}^{2}+\sigma_{1}\sigma_{2}\xi+\sigma_{2}^{2}.$
(6)

We see from this that anticorrelation $(\xi<0)$ leads to a net decrease in the
covariance function for a given stochastic process. This implies that a system
coupled to an anticorrelated environment will have a longer dephasing time as
compared to a system coupled to uncorrelated or completely correlated baths.

##  II Theory

Our theory is initialized by assuming that the total system can be separated
into system and reservoir variables such that

$\displaystyle H=H_{o}+\sum_{k}\hat{A}_{k}E_{k}(t)=H_{o}+H_{r}(t)$ (7)

where $H_{o}$ describes the system independent of reservoir with eigenstates
$H_{o}|\alpha\rangle=\hbar\omega_{\alpha}|\alpha\rangle$, $\hat{A}_{k}$ are a
set of quantum operators acting on the system subspace, and $E_{k}(t)$ are
stochastic variables representing the dynamics of the environment. Formally,
we write these in terms of an Ito stochastic differential equation of the form

$\displaystyle d{\mathbf{E}}$
$\displaystyle=\mathbf{A}(\tau,\mathbf{E}(\tau))dt+\mathbf{B}(\tau,\mathbf{E}_{\tau})\cdot
d\mathbf{W}_{t}$ (8)

where $\mathbf{W}$ is a vector of Wiener processes and
$\mathbf{A}(t,\mathbf{E})$ and $\mathbf{B}(t,\mathbf{E})$ define the the drift
and the diffusion. This general form allows for both nonlinear and geometric
processes to be incorporated into our model on an even footing. The process
$\mathbf{E}(t)$ is in general multidimensional and driven by a
multidimensional Wiener process with correlation matrix $\Sigma$. The process
itself can be written in integral form as

$\displaystyle\mathbf{E}(t)-\mathbf{E}({t_{o}})$
$\displaystyle=\int_{t_{o}}^{t}d\tau\mathbf{A}(\tau,\mathbf{E}(\tau))$
$\displaystyle+\int_{t_{o}}^{t}d\tau\mathbf{B}(\tau,\mathbf{E}(\tau))\cdot
d\mathbf{W}(t)$ (9)

with $dW_{i}(t)dW_{j}(t^{\prime})=\delta(t-t^{\prime})\Sigma_{ij}dt$ as the
generalized statement of Ito’s lemma. If we take the noise terms to be
correlated Ornstein-Uhlenbeck processes with

$\displaystyle
d\mathbf{E}_{t}=-\mathbf{A}\cdot\mathbf{E}_{t}dt+\mathbf{B}\cdot
d\mathbf{W}_{t}$ (10)

with $\mathbf{\Sigma}dt\delta(t-t^{\prime})=d\mathbf{W}\otimes d\mathbf{W}$ as
the correlation matrix, the general spectral density matrix takes the form

$\displaystyle
J(\omega)=\frac{1}{2\pi}(\mathbf{A}+i\omega)^{-1}\cdot\mathbf{B}\cdot\mathbf{\Sigma}\cdot\mathbf{B}^{T}\cdot(\mathbf{A}-i\omega)^{-1},$
(11)

as derived in Appendix A (c.f. Eq. 38). These terms enter into the quantum
dynamics of the reduced density matrix for the system variables via the Bloch
Redfield equations

$\displaystyle
d_{t}\rho_{\alpha\alpha^{\prime}}=-i(\omega_{\alpha}-\omega_{\alpha}^{\prime})\rho_{\alpha\alpha^{\prime}}-\sum_{\beta\beta^{\prime}}{\cal
R}_{\alpha\alpha^{\prime},\beta\beta^{\prime}}(\rho_{\beta\beta^{\prime}}-\rho_{\beta\beta^{\prime}}^{eq})$
(12)

where $\rho^{eq}$ is the equilibrium reduced density matrix and ${\cal R}$ is
the Bloch-Redfield tensor

$\displaystyle\begin{split}{\cal
R}_{\alpha\alpha^{\prime},\beta\beta^{\prime}}=\sum_{nm}\left\\{\delta_{\alpha^{\prime}\beta^{\prime}}\sum_{\gamma}J_{nm}(\omega_{\beta}-\omega_{\gamma})(A_{n})_{\gamma\beta}(A_{m})_{\alpha\gamma}\right.\\\
\left.-(J_{nm}(\omega_{\alpha}^{\prime}-\omega_{\beta}^{\prime})+J_{nm}(\omega_{\beta}-\omega_{\alpha}))(A_{n})_{\beta^{\prime}\alpha^{\prime}}(A_{m})_{\alpha\beta}\right.\\\
+\left.\delta_{\alpha\beta}\sum_{\gamma}J_{nm}(\omega_{\gamma}-\omega_{\beta}^{\prime})(A_{n})_{\beta^{\prime}\gamma}(A_{m})_{\gamma\alpha^{\prime}}\right\\}\end{split}$
(13)

where $(A_{n})_{\alpha\beta}=\langle\alpha|\hat{A}_{n}|\beta\rangle$ are the
matrix elements of the $\hat{A}_{n}$ operator in the eigenbasis of $H_{o}$ and
$J_{nm}(\omega)$ are elements of the generalized spectral matrix
characterizing the coupling between the system and its environment. [14, 15,
16, 17]

Figure 1: Sketch of 2-site model with correlated noise interactions.

Under the secular approximation in which the time evolution of the system is
slow compared to the characteristic correlation time of the environment
$|\omega_{\alpha\beta}-\omega_{\gamma\delta}|\ll 1/\tau_{c}$, the population
terms on the diagonal can be decoupled from the off-diagaonal coherence terms
via

$\displaystyle{\cal
R}_{ij;kl}^{sec}=\delta_{ij}\delta_{kl}+\delta_{ik}\delta_{jl}(1-\delta_{ij}\delta_{kl}).$
(14)

The time evolution of the reduced density matrix is strictly unitary under the
secular approximation, which guarantees that the ${\rm tr}(\rho)=1$ and all
diagonal elements are positive.

Table 1 gives a list of Redfield tensor elements for a single $SU(2)$ qubit
driven by correlated noise in both longitudinal ($\sigma_{z}$) and spin-
lattice $(\sigma_{x})$ terms. Within the secular approximation, the
longitudinal terms contribute to the pure dephasing $(T_{2})$ time, while the
spin-lattice term contributes to the relaxation. Even when the diffusion
matrix $\mathbf{B}$ is diagonal, cross-correlation enters the Redfield tensor
via non-vanishing terms involving the cross-spectral densities; however, these
terms only contribute to the non-secular terms of the tensor.

###  II.1 Coherence transfer between two qubits

We can easily generalise this model to accompany any number of states to
explore how correlated noise affects the relaxation dynamics of the system.
Here we consider a system of two spatially separated qubits, each driven by
locally correlated fields, coupled together by a static dipole-dipole
interaction and coupled to environment

$\displaystyle H$
$\displaystyle=\sum_{i=1,2}\frac{\epsilon_{i}}{2}\hat{\sigma}_{i}(3)+J(\hat{\sigma}_{1}^{+}\hat{\sigma}_{2}^{-}+\hat{\sigma}_{2}^{+}\hat{\sigma}_{1}^{-})+\sum_{j}\hat{A}_{j}E_{j}(t)$
(15)

where $E_{j}(t)$ are stochastic processes as above. Any system operators in
the state space $SU(2)\otimes SU(2)$ can be constructed by taking the tensor
products of $SU(2)$ Pauli matrices. Physically, this model could be achieved
in systems in which the energy of the local sites are strongly modulated by
the local phonon modes, as in the case of Jahn-Teller distortions of high-spin
octahedral $d^{4}$ coordination compounds where axial or equatorial
distortions split the otherwise degenerate $d_{z^{2}}$ and $d_{x^{2}-y^{2}}$
orbitals. Consequently, for a pair of octahedral sites, one can have symmetric
and antisymmetric combinations of normal modes that drive the Jahn-Teller
distortions of each site, giving rise to various degrees of correlation of the
thermal noise experienced at each metal site. Furthermore, it may be possible
through chemical or external stimulation to selectively enhance these modes.

If the energy states of the uncoupled qubits are identical,
$\epsilon_{1}=\epsilon_{2}$, then the two tunneling states are symmetric and
antisymmetric combinations of singly excited configurations $|10\rangle$ and
$|01\rangle$, that is,

$\displaystyle|\psi_{\pm}\rangle=\frac{1}{\sqrt{2}}(|01\rangle\pm|10\rangle)$
(16)

and dipole transitions from the lowest energy $|00\rangle$ are only to the
symmetric linear combination. If $J>0$ the symmetric state lies higher in
energy than the antisymmetric state and vice versa when $J<0$. For
$\epsilon_{1}\neq\epsilon_{2}$ both states are optically coupled to the ground
state, producing a pair of optical transitions, one of which being more
intense than the other (superradiant vs. subradiant).

We examine the effect of cross-correlation by computing the linear absorption
spectrum of the system for a suitable choice of parameters. From time-
dependent perturbation theory, the linear absorption spectrum is given by

$\displaystyle
S(\omega)\propto\left|\frac{1}{i\hbar}\int_{-\infty}^{\infty}e^{i\omega
t}\langle\mu(t)[\mu(0),\rho(-\infty)]]\rangle\right|^{2}$ (17)

where $\hat{\mu}(t)$ is the transition dipole operator in the
Heisenberg/Schrödinger representation at time $t$ and $\rho(-\infty)$ is the
system density matrix at $t\to-\infty$.

Figure 2: Linear response absorption spectra and associated line-width for a
pair of qubits subject to transverse (a,b) and longitudinal (c,d) noise terms
with various degrees of correlation.

Fig. 2(a-d) shows the linear absorption spectra and the corresponding relative
line widths for a pair of qubits with interaction $J/\epsilon=-0.2$ and with
correlation between either the two transverse or the two longitudinal noise
terms. Since only two noise terms are correlated, the spectral density matrix
is given by

$\displaystyle
J(z)=\frac{1}{2\pi}\left(\begin{array}[]{cc}\frac{2\xi\sigma_{12}\sigma_{1}+\sigma_{1}^{2}+\sigma_{12}^{2}}{\gamma_{1}^{2}+z^{2}}&\frac{\xi\sigma_{1}\sigma_{2}+\xi\sigma_{12}\sigma_{21}+\sigma_{12}\sigma_{2}+\sigma_{1}\sigma_{21}}{\left(z-i\gamma_{1}\right)\left(z+i\gamma_{2}\right)}\\\
\frac{\xi\sigma_{1}\sigma_{2}+\xi\sigma_{12}\sigma_{21}+\sigma_{12}\sigma_{2}+\sigma_{1}\sigma_{21}}{\left(z+i\gamma_{1}\right)\left(z-i\gamma_{2}\right)}&\frac{2\xi\sigma_{21}\sigma_{2}+\sigma_{2}^{2}+\sigma_{21}^{2}}{\gamma_{2}^{2}+z^{2}}\\\
\end{array}\right)$ (20)

to denote whether the term is local to site 1 or 2 or involves explicit
correlation between the two. Again, $\xi$ denotes whether or not the terms are
correlated or anticorrelated. In the transverse-transverse case, $J(z)$ is
evaluated at the transition frequency, since this coupling involves the
inelastic coupling to the environment; where as in the longitudinal-
longitudinal case, $J(z)$ is evaluated at $z=0$ since this corresponds to a
purely elastic coupling between the system and the environment.

Here, we see that correlations between transverse components have little
effect on the spectral line shape. We can understand this since the spectral
density terms are all evaluated at the transition frequency and are always
smaller than their longitudinal counterparts.

On the other hand, the correlations between longitudinal components have a
much more dramatic effect on both the transition intensity and line width,
with anti-correlated noise giving much sharper and more intense transitions.
We can understand this in the following way. According to the Kubo-Anderson
model, the spectral lineshape is determined by fluctuations in the transition
frequency. In the anticorrelated case, the local fluctuations are perfectly
synchronised but in opposite ways. That is, as the local site energy of one
increases, the other site energy always decreases. Therefore, the two local
fluctuations cancel each other out. In the fully correlated case, the
fluctuations are also perfectly synchronised, but both site energies increase
or decrease, which results in a broader spectral transition.

Figure 3: Purity(a,b) and fidelity (c,d) of composite $SU(2)\otimes SU(2)$
qubit pair vs time for systems prepared in a maximally entangled (Bell) state
corresponding to (a,c)$|\Phi^{+}\rangle$ or (b,d) $\Psi^{+}$.

We next consider the effect of initial-state preparation on the quantum
dynamics of the entangled qubits. For this, we introduce the following four
Bell-states

$\displaystyle|\Phi^{\pm}\rangle$
$\displaystyle=\frac{1}{2}(|00\rangle\pm|11\rangle)$ (21)
$\displaystyle|\Psi^{\pm}\rangle$
$\displaystyle=\frac{1}{2}(|01\rangle\pm|10\rangle)$ (22)

which correspond to the four maximally entangled quantum states of two qubits.
From the previous discussion, longitudinal correlations appear to have the
most profound effect on the dynamics, so we shall consider only that sort of
coupling in this example.

The purity, $\gamma={\rm tr}(\rho^{2})$, provides a useful measure of the
degree to which a quantum state is mixed. Mathematically, $\gamma=1$ for a
pure state since $\rho=\rho^{2}$ and takes a lower bound of $\gamma=1/4$
corresponding to the case where all 4 states of the system are equally
probable. Initially, the system is in a pure state with $\gamma=1$ and evolves
toward a mixed state as it evolves. At long time and low temperature, the
system will relax completely to the ground state $|00\rangle$ with $\gamma=1$.

Fig. 3(a,b) shows the purity of a composite $SU(2)\otimes SU(2)$ qubit pair vs
time for systems prepared in a maximally entangled (Bell) state and subject to
longitudinal noise with various degrees of correlation or anti-correlation. In
Fig. 3(a), we take the initial state as a coherence between the doubly excited
state $|11\rangle$ and the ground state $|00\rangle$, corresponding to the
$\Phi_{+}$ Bell state. Here, anticorrelation leads to a profound increase in
the system’s ability to retain its purity for nearly two orders of magnitude
in time longer than the fully correlated case. In contrast, if the initial
state is prepared in one of the $\Psi_{\pm}$ Bell states, corresponding to a
linear combination within the singly excited manifold of states, correlation
enhances the systems ability to retain purity. The only difference between the
two results is in the preparation of the initial state. This provides a
potentially useful experimental means for determining the correlation or anti-
correlation between local environments.

However, the fidelity

$\displaystyle F(\rho,\sigma)=\left({\rm
tr}\left(\sqrt{\sqrt{\rho}\sigma\sqrt{\rho}}\right)\right)^{2}$ (23)

is also an important consideration for whether or not a given Bell state is
suitable for an shared key. Fidelity provides a measure of the “closeness” of
two quantum states. It expresses the probability that one state will pass a
test to identify itself as the other. By symmetry,
$F(\rho,\sigma)=F(\sigma,\rho)$. In Fig. 3(c,d), we compute the time-evolved
fidelity $F(\rho_{0},\rho_{t})$ starting from the $\Phi^{+}$(c) or
$\Psi^{+}$(d) Bell states, versus various degrees of correlation between the
baths. For $\Phi^{+}$, the system loses fidelity rapidly and undergoes Rabi
oscillation within the double excitation manifold spanned by $\Phi^{+}$ and
$\Phi^{-}$. The fidelity eventually relaxes to $F=1/2$ for a long time,
corresponding to complete relaxation into the ground state $|00\rangle$. As
with purity, the envelope of fidelity is enhanced by anti-correlated noise.

In contrast, the correlated noise helps to maintain both the purity and
fidelity of the $\Psi^{+}$ Bell state. This state is an eigenstate of the bare
system Hamiltonian which can be prepared by direct photoexcitation from the
ground state. It is curious that the above results suggest that correlated
noise suppresses the optical response. However, the optical response is
actually a measure of the coherence between the ground state and $\Psi^{+}$
and not a measure of the purity or fidelity of a given state.

##  III Discussion

In this paper we explored the role of noise correlation on a model open
quantum system consisting of one and two coupled $SU(2)$ qubits and showed how
the dynamics and spectroscopy of the system can be profoundly affected by
environmental correlations. This has deep implications for searching materials
suitable for quantum communications and computation applications in which long
coherence times and retention of are required. In the case of super-dense
coding, a sender (A) and a receiver (B) can communicate a number of classical
bits of information by only transmitting a smaller number of qubits, provided
that A and B are presharing an entangled resource. [18] Since this resource is
subject to environmental noise, the ability of A and B to perform super-dense
coding hinges on their ability to maintain the fidelity of the state of the
shared resource. Similarly, quantum teleportation requires the sender and
receiver to share a maximally entangled state. [19] Our results suggest that
by knowing whether the state is subject to correlated or anticorrelated noise,
A and B can be ensured that their shared resource state can maintain its
purity long enough for the information to be communicated. We also suggest
that the local noise correlation can be tuned by manipulating the local
environment around the two qubits.

###### Acknowledgements.

The work at the University of Houston was funded in part by the National
Science Foundation (CHE-2102506) and the Robert A. Welch Foundation (E-1337).
The work at Los Alamos National Laboratory was funded by the Laboratory
Directed Research and Development (LDRD) programme, 20220047DR. The work at
Georgia Tech was funded by the National Science Foundation (DMR-1904293).

## Author Contributions

Eric Bittner: Supervision, Funding acquisition, Conceptualisation,
Methodology, Formal analysis, Validation, Writing; Hao Li: Formal analysis,
Methodology, Validation; Syad A Shah: Conceptualisation; Carlos Silva:
Conceptualisation Funding acquisition; Andrei Piryatinski: Conceptualisation,
Validation, Funding acquisition. All authors contributed to the final draft
and editing of this manuscript.

## Data Availability

Data supporting the findings of this study are available from the
corresponding author on a reasonable request.

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##  Appendix A Correlations amongst random variables

The Ornstein-Uhlenbeck process is a very useful method to account for many
Markovian stochastic processes. Its multivariate representation is even more
practical for physical processes. Here we discuss the multivariate Ornstein-
Uhlenbeck process, including correlated Wiener processes, for the purpose of
tackling realistic physical problems such as chromorphores coupled to their
respective phonon environments but interacting with a common bath.

We write the multivariate Ornstein-Uhlenbeck process as a vector ${\bm{E}}(t)$
composed of individual processes $X_{i}(t)$. The stochastic differential
equation reads

${\rm d}{\bm{E}}(t)=A\left[{\bm{\mu}}-{\bm{E}}(t)\right]{\rm d}t+B{\rm
d}{\bm{W}}(t),$ (24)

in which $A$ and $B$ are coefficient matrices, ${\bm{\mu}}$ is the vector of
the Wiener process drift $\mu_{i}$ corresponding to $W_{i}$. ${\bm{W}}(t)$ is
the vector of Wiener processes $W_{i}(t)$ which are correlated through the
correlation matrix

$\xi(t,t^{\prime})\equiv\delta_{tt^{\prime}}{\rm d}{\bm{W}}(t){\rm
d}{\bm{W}}(t^{\prime})^{\rm T}/{\rm d}t,$ (25)

where the angular brackets represent the ensemble average. The matrix elements
$\xi_{ij}={\rm d}W_{i}(t){\rm d}W_{j}(t)/{\rm d}t$ are defined through the Itô
isometry in higher dimensions. Obviously $\xi_{ii}=1$ according to the
quadratic variation $({\rm d}W_{t})^{2}={\rm d}t$. $\xi_{ij}$ varies from -1
to 1, respectively, corresponding to the fully anticorrelated and fully
correlated cases. $\xi_{ij}=0$ means that the two Wiener processes are
completely uncorrelated.

According to the Itô’s lemma, one finds the solution

${\bm{E}}(t)=e^{-At}{\bm{E}}(0)+(\mathbb{1}-e^{-At}){\bm{\mu}}+\int_{0}^{t}e^{-A(t-t^{\prime})}B{\rm
d}{\bm{W}}(t^{\prime}),$ (26)

where ${\bm{E}}(0)$ is the initial condition of the process ${\bm{E}}(t)$, the
mean value

$\langle{\bm{E}}(t)\rangle=e^{-At}\langle{\bm{E}}(0)\rangle+(\mathbb{1}-e^{-At}){\bm{\mu}},$
(27)

and the correlation function

$\displaystyle\left<{\bm{E}}(t),{\bm{E}}^{\rm T}(s)\right>$
$\displaystyle\equiv\left<\left[{\bm{E}}(t)-\langle{\bm{E}}(t)\rangle\right]\left[{\bm{E}}(s)-\langle{\bm{E}}(s)\rangle\right]^{\rm
T}\right>$ $\displaystyle=e^{-At}\left<{\bm{E}}(0),{\bm{E}}^{\rm
T}(0)\right>e^{-A^{\rm T}s}+\int_{0}^{\min(s,t)}e^{-A(t-t^{\prime})}B\xi
B^{\rm T}e^{-A^{\rm T}(s-t^{\prime})}{\rm d}t^{\prime}$ (28)

following the Itô isometry in higher dimensions.

If $AA^{\rm T}=A^{\rm T}A$, one can find a unitary matrix $S$ to diagonalize
the coefficient matrix $SAS^{\dagger}=SA^{\rm T}S^{\dagger}={\rm
diag}(\gamma_{1},\gamma_{2},\dots,\gamma_{n})$. For deterministic initial
condition $\left<{\bm{E}}(0),{\bm{E}}^{\rm T}(0)\right>=0$, so does the
correlation function $\left<{\bm{E}}(t),{\bm{E}}^{\rm
T}(s)\right>=S^{\dagger}G(t,s)S$, in which

$\displaystyle\begin{split}\left[G(t,s)\right]_{ij}&=\frac{\left(B\xi B^{\rm
T}\right)_{ij}}{\gamma_{i}+\gamma_{j}}\left[e^{-\gamma_{i}|t-s|}-e^{-\gamma_{i}t-\gamma_{j}s}\right]\qquad(t\geq
s),\\\ \left[G(t,s)\right]_{ij}&=\frac{\left(B\xi B^{\rm
T}\right)_{ij}}{\gamma_{i}+\gamma_{j}}\left[e^{-\gamma_{j}|t-s|}-e^{-\gamma_{i}t-\gamma_{j}s}\right]\qquad(t\leq
s).\end{split}$ (29)

If the real parts of all $A$’s eigenvalues are positive, one finds the
stationary solution

${\bm{E}}_{\rm s}(t)={\bm{\mu}}+\int_{-\infty}^{t}e^{-A(t-t^{\prime})}B{\rm
d}{\bm{W}}(t^{\prime}),$ (30)

with the stationary correlation matrix

$\left<{\bm{E}}_{\rm s}(t),{\bm{E}}_{\rm s}^{\rm
T}(s)\right>=\int_{-\infty}^{\min(s,t)}e^{-A(t-t^{\prime})}B\xi B^{\rm
T}e^{-A^{\rm T}(s-t^{\prime})}{\rm d}{\bm{W}}(t^{\prime}).$ (31)

We define the stationary covariance matrix

$\sigma=\left<{\bm{E}}_{\rm s}(t),{\bm{E}}_{\rm s}^{\rm T}(t)\right>,$ (32)

then find a useful algebraic equation for stationary covariance matrix

$A\sigma+\sigma A^{\rm T}=B\xi B^{\rm T}.$ (33)

For $s<t$ the stationary correlation function Eq.(31) can be written as

$\displaystyle\left<{\bm{E}}_{\rm s}(t),{\bm{E}}_{\rm s}^{\rm T}(s)\right>$
$\displaystyle=e^{-A(t-s)}\int_{-\infty}^{s}e^{-A(s-t^{\prime})}B\xi B^{\rm
T}e^{-A^{\rm T}(s-t^{\prime})}{\rm d}t^{\prime}$
$\displaystyle=e^{-A(t-s)}\sigma\qquad s<t,$ (34)

and

$=\sigma e^{-A^{\rm T}(s-t)}\qquad s>t.$ (35)

The correlation function only depends on the time difference $|t-s|$ as
expected for the stationary solution. We define the stationary correlation
matrix $G_{\rm s}(\tau)=\left<{\bm{E}}_{\rm s}(t),{\bm{E}}_{\rm s}^{\rm
T}(t-\tau)\right>$, obviously $G_{\rm s}(0)=\sigma$. Then the above relation
can be written in the form of the regression theorem

$\displaystyle\frac{\rm d}{\rm d\tau}\left[G_{\rm s}(\tau)\right]=\frac{\rm
d}{\rm d\tau}\left<{\bm{E}}_{\rm s}(\tau),{\bm{E}}_{\rm s}^{\rm
T}(0)\right>=-AG_{\rm s}(\tau).$ (36)

Noting that $G_{\rm s}(0)=\sigma$, one can compute the stationary correlation
matrix.

Since $\sigma^{\rm T}=\sigma$, we have

$G_{\rm s}(\tau)=\left[G_{\rm s}(-\tau)\right]^{\rm T}.$ (37)

Therefore, one can find the spectrum matrix as the Fourier transform of the
autocorrelation matrix $G_{\rm s}(\tau)$

$\displaystyle J(\omega)$
$\displaystyle=\frac{1}{2\pi}\int_{-\infty}^{\infty}e^{-i\omega\tau}G_{\rm
s}(\tau){\rm d}\tau$
$\displaystyle=\frac{1}{2\pi}\left(A+i\omega\right)^{-1}B\xi B^{\rm
T}\left(A-i\omega\right)^{-1}.$ (38)

As an example, we consider the case of the case of two correlated modes, in
which we define the 2D Ornstein-Uhlenbeck process by the SDEs

$\displaystyle{\rm d}E_{1}(t)$ $\displaystyle=-\gamma_{1}E_{1}(t){\rm
d}t+\sigma_{11}{\rm d}B_{1}(t)+\sigma_{12}{\rm d}B_{2}(t),$ $\displaystyle{\rm
d}E_{2}(t)$ $\displaystyle=-\gamma_{2}E_{2}(t){\rm d}t+\sigma_{21}{\rm
d}B_{1}(t)+\sigma_{22}{\rm d}B_{2}(t).$

The two Wiener processes $B_{1}(t)$ and $B_{2}(t)$ are coupled through the
correlation parameter $\xi={\rm d}B_{1}(t){\rm d}B_{2}(t)/{\rm d}t$. The range
of $\xi$ is between $-1$ to $1$ corresponding to the cases of complete anti-
correlation and correlation, respectively. $\xi=0$ means that the two Wiener
processes are completely decoupled. The solutions of the OU processes are

$\displaystyle E_{1}(t)$
$\displaystyle=e^{-\gamma_{1}t}E_{1}(0)+\sigma_{11}\int_{0}^{t}e^{-\gamma_{1}(t-s)}{\rm
d}B_{1}(s)+\sigma_{12}\int_{0}^{t}e^{-\gamma_{1}(t-s)}{\rm d}B_{2}(s),$
$\displaystyle E_{2}(t)$
$\displaystyle=e^{-\gamma_{2}t}E_{2}(0)+\sigma_{21}\int_{0}^{t}e^{-\gamma_{2}(t-s)}{\rm
d}B_{1}(s)+\sigma_{22}\int_{0}^{t}e^{-\gamma_{2}(t-s)}{\rm d}B_{2}(s).$

From this we compute the mean values

$\displaystyle\left<E_{1}(t)\right>=\left<E_{1}(0)\right>e^{-\gamma_{1}t},$
$\displaystyle\left<E_{2}(t)\right>=\left<E_{2}(0)\right>e^{-\gamma_{2}t},$

as well as the correlation functions

$\displaystyle{\rm Cov}\left[E_{1}(t),E_{1}(s)\right]$
$\displaystyle=\left<E_{1}(0)^{2}\right>e^{-\gamma_{1}(t+s)}+\frac{\sigma_{11}^{2}+\sigma_{12}^{2}+2\xi\sigma_{11}\sigma_{12}}{2\gamma_{1}}\left[e^{-\gamma_{1}|t-s|}-e^{-\gamma_{1}(t+s)}\right],$
$\displaystyle{\rm Cov}\left[E_{2}(t),E_{2}(s)\right]$
$\displaystyle=\left<E_{2}(0)^{2}\right>e^{-\gamma_{2}(t+s)}+\frac{\sigma_{21}^{2}+\sigma_{22}^{2}+2\xi\sigma_{21}\sigma_{22}}{2\gamma_{2}}\left[e^{-\gamma_{2}|t-s|}-e^{-\gamma_{2}(t+s)}\right],$
$\displaystyle{\rm Cov}\left[E_{1}(t),E_{2}(s)\right]$
$\displaystyle=\left<E_{1}(0),E_{2}(0)\right>e^{-\gamma_{1}t-\gamma_{2}s}+\frac{\sigma_{11}\sigma_{21}+\sigma_{12}\sigma_{22}+\xi\sigma_{11}\sigma_{22}+\xi\sigma_{12}\sigma_{21}}{\gamma_{1}+\gamma_{2}}e^{-\gamma_{1}t-\gamma_{2}s}\left[e^{(\gamma_{1}+\gamma_{2})\min(s,t)}-1\right],$
$\displaystyle{\rm Cov}\left[E_{2}(t),E_{1}(s)\right]$
$\displaystyle=\left<E_{1}(0),E_{2}(0)\right>e^{-\gamma_{2}t-\gamma_{1}s}+\frac{\sigma_{11}\sigma_{21}+\sigma_{12}\sigma_{22}+\xi\sigma_{11}\sigma_{22}+\xi\sigma_{12}\sigma_{21}}{\gamma_{1}+\gamma_{2}}e^{-\gamma_{2}t-\gamma_{1}s}\left[e^{(\gamma_{1}+\gamma_{2})\min(s,t)}-1\right].$

Using these we find the spectral density matrix for the correlated processes
as

$J(\omega)=\frac{1}{2\pi}\begin{bmatrix}\frac{\sigma_{11}^{2}+2\xi\sigma_{11}\sigma_{22}+\sigma_{12}^{2}}{\gamma_{1}^{2}+\omega^{2}}&\frac{\sigma_{12}\sigma_{22}+\sigma_{11}\sigma_{21}+\xi(\sigma_{12}\sigma_{21}+\sigma_{11}\sigma_{22})}{(\gamma_{1}+i\omega)(\gamma_{2}-i\omega)}\\\
\frac{\sigma_{12}\sigma_{22}+\sigma_{11}\sigma_{21}+\xi(\sigma_{12}\sigma_{21}+\sigma_{11}\sigma_{22})}{(\gamma_{1}-i\omega)(\gamma_{2}+i\omega)}&\frac{\sigma_{21}^{2}+2\xi\sigma_{21}\sigma_{22}+\sigma_{22}^{2}}{\gamma_{2}^{2}+\omega^{2}}\end{bmatrix}.$
(39)

##  Appendix B Redfield tensor elements for cross correlation between $x$ and
$z$ for a single $SU(2)$ qubit

.

The Bloch-Redfield equations give the quantum dynamics of the reduced density
matrix according to

$\displaystyle
d_{t}\rho_{\alpha\alpha^{\prime}}=-i(\omega_{\alpha}-\omega_{\alpha}^{\prime})\rho_{\alpha\alpha^{\prime}}-\sum_{\beta\beta^{\prime}}{\cal
R}_{\alpha\alpha^{\prime};\beta\beta^{\prime}}(\rho_{\beta\beta^{\prime}}-\rho_{beta\beta^{\prime}}^{eq}$
(40)

where $\rho^{eq}$ is the equilibrium reduced density matrix and ${\cal R}$ is
the Bloch-Redfield tensor with elements [14, 15, 16, 17]

$\displaystyle\begin{split}{\cal
R}_{\alpha\alpha^{\prime};\beta\beta^{\prime}}=\sum_{nm}\left\\{\delta_{\alpha^{\prime}\beta^{\prime}}\sum_{\gamma}J_{nm}(\omega_{\beta}-\omega_{\gamma})(A_{n})_{\gamma\beta}(A_{m})_{\alpha\gamma}\right.\\\
\left.-(J_{nm}(\omega_{\alpha}^{\prime}-\omega_{\beta}^{\prime})+J_{nm}(\omega_{\beta}-\omega_{\alpha}))(A_{n})_{\beta^{\prime}\alpha^{\prime}}(A_{m})_{\alpha\beta}\right.\\\
+\left.\delta_{\alpha\beta}\sum_{\gamma}J_{nm}(\omega_{\gamma}-\omega_{\beta}^{\prime})(A_{n})_{\beta^{\prime}\gamma}(A_{m})_{\gamma\alpha^{\prime}}\right\\}\end{split}$
(41)

where $(A_{n})_{\alpha\beta}=\langle\alpha|\hat{A}_{n}|\beta\rangle$ are the
matrix elements of the $\hat{A}_{n}$ operator in the eigenbasis of $H_{o}$ and
$J_{nm}(\omega)$ are elements of the generalized spectral matrix
characterizing the coupling between the system and its environment. Table 1
gives the tensor elements for the case of a single qubit with transition
frequency $\epsilon$ coupled to a noisy environment through both longitudinal
(through $\hat{\sigma}_{z}$) and transverse (through$\hat{\sigma}_{x}$ or
$\hat{\sigma}_{y}$). $J_{ij}(\omega)$ to denote the spectral density
associated with the correlation function $\langle
E_{i}(t)E_{j}(t^{\prime})\rangle$.

The second column indicates whether or not $R_{ijkl}$ is non-vanishing within
the secular approximation.

$\displaystyle
R^{sec}_{ij;kl}=\left(\delta_{ij}\delta_{kl}+\delta_{ik}\delta_{jl}(1-\delta_{ij}\delta_{kl})\right)R_{ij;kl}$
(42)

When operating under this limit, the system populations are decoupled from the
coherences, following a regular Pauli Master equation with the population rate
matrix $R_{iikk}$. This ensures population conservation and achieves the
correct thermal equilibrium over extended periods. Under this approximation,
the density matrix exhibits the appropriate physical behaviour with ${\rm
Tr}[\rho]=1$. The population rate matrix, being real, facilitates exponential
relaxation of the populations. Coherences are also fully separated from the
population and experience attenuation by the dephasing rates $R_{ij;ij}$.
Generally, $R_{ij;ij}$ is complex, with its imaginary component representing
bath-induced energy shifts. For example, under the secular approximation, we
expect that $R_{11;11}+R_{22;11}=R_{22;22}+R_{11;22}=0$ and
$R_{12;12}=R_{21;21}^{*}$ for the coherence terms. The presence of the cross-
correlation terms does not lead to a violation of these conditions.

$\displaystyle\begin{array}[]{ccccl|l|l}i&j&k&l&\text{Secular}&{\cal
R}_{ij;kl}&\text{Ornstein-Uhlenbeck}\\\ \hline\cr
1&1&1&1&\text{Secular}&J_{\text{xx}}(-\epsilon)+J_{\text{xx}}(\epsilon)&\frac{2s_{x}\left(s_{x}+2\xi
s_{\text{xz}}\right)}{\gamma_{x}^{2}+\epsilon^{2}}\\\ 1&1&1&2&\text{Non-
Secular}&-J_{\text{xz}}(0)-J_{\text{zx}}(0)&-\frac{2\left(s_{x}\left(s_{\text{xz}}+\xi
s_{z}\right)+\xi s_{\text{xz}}^{2}\right)}{\gamma_{x}\gamma_{z}}\\\
1&1&2&1&\text{Non-
Secular}&J_{\text{xz}}(-\epsilon)-J_{\text{zx}}(-\epsilon)-2J_{\text{zx}}(0)&-\frac{2\left(\gamma_{x}^{2}\left(i\epsilon\gamma_{z}+\gamma_{z}^{2}+\epsilon^{2}\right)-i\epsilon\gamma_{x}\gamma_{z}^{2}+\epsilon^{2}\left(\gamma_{z}^{2}+\epsilon^{2}\right)\right)\left(s_{x}\left(s_{\text{xz}}+\xi
s_{z}\right)+\xi
s_{\text{xz}}^{2}\right)}{\gamma_{x}\gamma_{z}\left(\gamma_{x}^{2}+\epsilon^{2}\right)\left(\gamma_{z}^{2}+\epsilon^{2}\right)}\\\
1&1&2&2&\text{Secular}&-J_{\text{xx}}(-\epsilon)-J_{\text{xx}}(\epsilon)&-\frac{2s_{x}\left(s_{x}+2\xi
s_{\text{xz}}\right)}{\gamma_{x}^{2}+\epsilon^{2}}\\\ 1&2&1&1&\text{Non-
Secular}&-2J_{\text{xz}}(-\epsilon)-J_{\text{xz}}(0)+J_{\text{zx}}(0)&-\frac{2\left(s_{x}\left(s_{\text{xz}}+\xi
s_{z}\right)+\xi
s_{\text{xz}}^{2}\right)}{\left(\epsilon+i\gamma_{x}\right)\left(\epsilon-i\gamma_{z}\right)}\\\
1&2&1&2&\text{Secular}&2\left(J_{\text{xx}}(\epsilon)+2J_{\text{zz}}(0)\right)&\frac{4s_{\text{xz}}^{2}\left(\gamma_{x}^{2}+\epsilon^{2}\right)+4\xi
s_{\text{xz}}\left(2s_{z}\left(\gamma_{x}^{2}+\epsilon^{2}\right)+s_{x}\gamma_{z}^{2}\right)+2s_{x}^{2}\gamma_{z}^{2}}{\gamma_{z}^{2}\left(\gamma_{x}^{2}+\epsilon^{2}\right)}\\\
1&2&2&1&\text{Non-
Secular}&-2J_{\text{xx}}(-\epsilon)&-\frac{2s_{x}\left(s_{x}+2\xi
s_{\text{xz}}\right)}{\gamma_{x}^{2}+\epsilon^{2}}\\\ 1&2&2&2&\text{Non-
Secular}&J_{\text{xz}}(-\epsilon)+J_{\text{zx}}(-\epsilon)&\frac{2\left(\gamma_{x}\gamma_{z}+\epsilon^{2}\right)\left(s_{x}\left(s_{\text{xz}}+\xi
s_{z}\right)+\xi
s_{\text{xz}}^{2}\right)}{\left(\gamma_{x}^{2}+\epsilon^{2}\right)\left(\gamma_{z}^{2}+\epsilon^{2}\right)}\\\
2&1&1&1&\text{Non-
Secular}&-J_{\text{xz}}(\epsilon)-J_{\text{zx}}(\epsilon)&-\frac{2\left(\gamma_{x}\gamma_{z}+\epsilon^{2}\right)\left(s_{x}\left(s_{\text{xz}}+\xi
s_{z}\right)+\xi
s_{\text{xz}}^{2}\right)}{\left(\gamma_{x}^{2}+\epsilon^{2}\right)\left(\gamma_{z}^{2}+\epsilon^{2}\right)}\\\
2&1&1&2&\text{Non-
Secular}&-2J_{\text{xx}}(\epsilon)&-\frac{2s_{x}\left(s_{x}+2\xi
s_{\text{xz}}\right)}{\gamma_{x}^{2}+\epsilon^{2}}\\\
2&1&2&1&\text{Secular}&2\left(J_{\text{xx}}(-\epsilon)+2J_{\text{zz}}(0)\right)&\frac{4s_{\text{xz}}^{2}\left(\gamma_{x}^{2}+\epsilon^{2}\right)+4\xi
s_{\text{xz}}\left(2s_{z}\left(\gamma_{x}^{2}+\epsilon^{2}\right)+s_{x}\gamma_{z}^{2}\right)+2s_{x}^{2}\gamma_{z}^{2}}{\gamma_{z}^{2}\left(\gamma_{x}^{2}+\epsilon^{2}\right)}\\\
2&1&2&2&\text{Non-
Secular}&2J_{\text{xz}}(\epsilon)+J_{\text{xz}}(0)-J_{\text{zx}}(0)&\frac{2\left(s_{x}\left(s_{\text{xz}}+\xi
s_{z}\right)+\xi
s_{\text{xz}}^{2}\right)}{\left(\epsilon-i\gamma_{x}\right)\left(\epsilon+i\gamma_{z}\right)}\\\
2&2&1&1&\text{Secular}&-J_{\text{xx}}(-\epsilon)-J_{\text{xx}}(\epsilon)&-\frac{2s_{x}\left(s_{x}+2\xi
s_{\text{xz}}\right)}{\gamma_{x}^{2}+\epsilon^{2}}\\\ 2&2&1&2&\text{Non-
Secular}&-J_{\text{xz}}(\epsilon)+J_{\text{zx}}(\epsilon)+2J_{\text{zx}}(0)&\frac{2\left(\gamma_{x}^{2}\left(-i\epsilon\gamma_{z}+\gamma_{z}^{2}+\epsilon^{2}\right)+i\epsilon\gamma_{x}\gamma_{z}^{2}+\epsilon^{2}\left(\gamma_{z}^{2}+\epsilon^{2}\right)\right)\left(s_{x}\left(s_{\text{xz}}+\xi
s_{z}\right)+\xi
s_{\text{xz}}^{2}\right)}{\gamma_{x}\gamma_{z}\left(\gamma_{x}^{2}+\epsilon^{2}\right)\left(\gamma_{z}^{2}+\epsilon^{2}\right)}\\\
2&2&2&1&\text{Non-
Secular}&J_{\text{xz}}(0)+J_{\text{zx}}(0)&\frac{2\left(s_{x}\left(s_{\text{xz}}+\xi
s_{z}\right)+\xi s_{\text{xz}}^{2}\right)}{\gamma_{x}\gamma_{z}}\\\
2&2&2&2&\text{Secular}&J_{\text{xx}}(-\epsilon)+J_{\text{xx}}(\epsilon)&\frac{2s_{x}\left(s_{x}+2\xi
s_{\text{xz}}\right)}{\gamma_{x}^{2}+\epsilon^{2}}\\\ \hline\cr\end{array}$
(60) Table 1: Redfield tensor elements for $SU(2)$ qubit with cross-
correlation between $\hat{\sigma}(1)$ and $\hat{\sigma}(3)$ noise terms. The
second column indicates whether the term survives under the secular
approximation, which separates the evolution of the population and the
coherence terms. The last column gives the tensor element within the Ornstein-
Uhlenbeck model.


# A value distribution result related to Hayman’s alternative

Kuldeep Singh Charak  Kuldeep Singh Charak Department of Mathematics
University of Jammu Jammu-180 006 India<EMAIL_ADDRESS>and  Anil
Singh  Anil Singh Department of Mathematics University of Jammu Jammu-180 006
India<EMAIL_ADDRESS>
###### Abstract.

Motivated by Bloch’s Principle, we prove a value distribution result for
meromorphic functions which is related to Hayman’s alternative in certain
sense.

††footnotetext: 2010 Mathematics Subject Classification. 30D35,
30D45.††footnotetext: Keywords and phrases. Meromorphic function, Value
distribution theory, Normal families, Bloch’s principle.

##  1\. Introduction and Main Result

The reader is assumed to be familiar with the standard notations of Nevanlinna
value distribution theory of meromorphic functions (one may refer to [4, 6])
such as $T\left(r,f\right)$, $m\left(r,f\right)$, $N\left(r,f\right),$ etc. We
shall denote the class of all meromorphic functions on a domain $D$ in
$\mathbb{C}$ by $\mathcal{M}(D)$ and we shall write, ‘$\left\langle
f,D\right\rangle\in\mathcal{P}$’ for ‘$f\in\mathcal{M}(D)$ satisfies the
property $\mathcal{P}$ on $D$’ .

We say that $\phi\in\mathcal{M}(\mathbb{C})$ is a small function of
$f\in\mathcal{M}(\mathbb{C})$ if $T(r,\phi)=S(r,f)$ as $r\to\infty$ possibly
outside a set of $r$ of finite linear measure.

W.K. Hayman proved the following ‘Picard type’ theorem, also known as Hayman’s
alternative:

######  Theorem 1.1.

[7] Let $f\in\mathcal{M}(\mathbb{C})$ and let $l\geq 1.$ Suppose that
$f(z)\neq 0,\mbox{ and }f^{(l)}(z)-1\neq 0$ for all $z\in\mathbb{C}.$ Then $f$
is constant.

A subfamily $\mathcal{F}$ of $\mathcal{M}(D)$ is said to be normal in $D$ if
every sequence of members of $\mathcal{F}$ contains a subsequence that
converges locally uniformly (w.r.t. the spherical metric) in $D$. Recall
Bloch’s Principle (see[10, 11]): A subfamily $\mathcal{F}$ of $\mathcal{M}(D)$
with $\left\langle f,D\right\rangle\in\mathcal{P}$ for each $f\in\mathcal{F}$
is likely to be normal on $D$ if $\mathcal{P}$ reduces every
$f\in\mathcal{M}(\mathbb{C})$ to a constant. Neither Bloch’s Principle nor its
converse is true (see [1, 2, 3, 8, 10]).

According to Bloch’s Principle, to every ‘Picard type’ theorem there
corresponds a normality criterion. A normality criterion corresponding to
Theorem 1.1 was proved by Y.Gu as follows:

######  Theorem 1.2.

[5] Let $\mathcal{F}\subseteq\mathcal{M}(D)$ and let $l\geq 1.$ Suppose that
$f(z)\neq 0,\mbox{ and }f^{(l)}(z)-1\neq 0$ for all $z\in D$ and
$f\in\mathcal{F}.$ Then $\mathcal{F}$ is normal in $D.$

The constants $0\mbox{ and }1$ in Theorem 1.1 and Theorem 1.2 can be replaced
by arbitrary constants $a\mbox{ and }b\neq 0$:

######  Theorem 1.3.

[7] Let $f\in\mathcal{M}(D)$ and let $l\geq 1.$ Suppose that $f(z)\neq
a,\mbox{ and }f^{(l)}(z)-b\neq 0$ for all $z\in\mathbb{C},$ where
$a,b\in\mathbb{C},~{}b\neq 0.$ Then $f$ is constant.

######  Theorem 1.4.

[5] Let $\mathcal{F}\subseteq\mathcal{M}(D)$ and let $l\geq 1.$ Suppose that
$f(z)\neq a,\mbox{ and }f^{(l)}(z)-b\neq 0$ for all $z\in D,$
$f\in\mathcal{F}$, where $a,b\in\mathbb{C}$ with $~{}b\neq 0.$ Then
$\mathcal{F}$ is normal in $D.$

Note that if $l\geq 1$ and $b\in\mathbb{C}\setminus\left\\{0\right\\}$, then
there is a polynomial $P(z)$ such that $P^{(l)}(z)=b.$ Using this observation,
Theorem 1.3 and Theorem 1.4 can be restated as:

######  Theorem 1.5.

Suppose that $P(z)$ is a polynomial of degree $l\geq 1$ and $a\in\mathbb{C}.$
If $f\in\mathcal{M}(\mathbb{C})$ is such that $f(z)\neq a\mbox{ and
}\left(f(z)-P(z)\right)^{(l)}\neq 0,$ then $f$ is constant.

######  Theorem 1.6.

Suppose that $P(z)$ is a polynomial of degree $l\geq 1$ and $a\in\mathbb{C}.$
If $\mathcal{F}\subseteq\mathcal{M}(D)$ is such that each $f\in\mathcal{F}$
satisfies:

$f(z)\neq a\mbox{ and }\left(f(z)-P(z)\right)^{(l)}\neq 0,$

then $\mathcal{F}$ is normal in $D.$

######  Remark 1.7.

Put $g=f-P\mbox{ and }R=Q-P$, where $P\mbox{ and }Q$ are polynomials with
$deg(P-Q)=deg(Q)=l$ and $Q$ is non-constant. If $f(z)-P(z)\neq 0$ and
$\left(f(z)-Q(z)\right)^{(l)}\neq 0,$ then by using Theorem 1.5, we find that
$f(z)=P(z)+c$, for some constant $c\neq 0.$

Remark 1.7 shows that Theorem 1.3 does not hold if $a$ is replaced by some
non-constant function.

######  Remark 1.8.

Suppose $f$ is an entire function such that $f-g$ has only finitely many zeros
in the plane, where $g$ is some non-constant entire function. Further, let

$F(z)=\sum_{k=1}^{n}a_{k}(z)\left(f-g\right)^{(k)}$

omits $1,$ where $a_{k}(z)$ are small functions of $f.$ Then by using Theorem
3.2 in [6], we find that $f(z)=g(z)+p(z)$, for some polynomial $p(z).$ Indeed,

$T(r,f-g)<\overline{N}(r,f-g)+N(r,{1\over f-g})+\overline{N}\left(r,{1\over
F-1}\right)-N_{0}\left(r,{1\over F^{\prime}}\right)+S(r,f),$

where $N_{0}\left(r,{1\over F^{\prime}}\right)$ is the counting function of
the zeros of $F^{\prime}$ which are not zeros of $F-1.$

Since $f-g$ is entire and has only finitely many zeros, it follows that

$\displaystyle T(r,f-g)<\overline{N}\left(r,{1\over
F-1}\right)-N_{0}\left(r,{1\over F^{\prime}}\right)+S(r,f)$
$\displaystyle\Rightarrow T(r,f-g)<\overline{N}\left(r,{1\over
F-1}\right)+S(r,f).$

If $f-g$ is transcendental, then $F(z)=1$ must have infinitely many roots,
which is a contradiction and hence $f-g$ must be a polynomial, say $p(z);$
that is, $f(z)=g(z)+p(z).$

Let $P\mbox{ and }Q$ be polynomials with $1\leq deg(P)<deg(Q)=l$ and
$\mathcal{P}$ be the property defined as follows: $``\left\langle
f,D\right\rangle\in\mathcal{P}\Leftrightarrow f-P\neq 0\mbox{ and
}\left(f-Q\right)^{(l)}\neq 0\mbox{ on }D$”. That is, $f$ satisfies the
property $\mathcal{P}$ if, and only if $f-P\mbox{ and }\left(f-Q\right)^{(l)}$
have no zeros in $D.$ With this $\mathcal{P}$, Theorem 1.6 immediately yields:

######  Theorem 1.9.

The family $\mathcal{F}:=\left\\{f\in\mathcal{M}\left(D\right):\left\langle
f,D\right\rangle\in\mathcal{P}\right\\}$ is normal in $D.$

Note that Remark 1.7 and Theorem 1.9 provide a counterexample to the converse
of Bloch’s Principle.

W. Schwick generalized Theorem 1.2 :

######  Theorem 1.10.

[12] Let $g\not\equiv 0$ be in $\mathcal{M}(D)$ and let $l\in\mathbb{N}.$ Let
$\mathcal{F}\subseteq\mathcal{M}(D)$ be such that $f\neq 0,$ $f^{(l)}\neq g,$
and $f\mbox{ and }g$ have no common poles for each $f\in\mathcal{F}.$ Then
$\mathcal{F}$ is normal in $D.$

According to the converse of Bloch’s Principle, one may find a ‘Picard type’
theorem corresponding to Theorem 1.10, and this is the purpose of this paper.
In fact, we prove the following value distribution result corresponding to
Theorem 1.10 which is related to Hayman’s alternative in certain sense:

######  Theorem 1.11.

Suppose that $f\in\mathcal{M}(\mathbb{C})$ is transcendental and $\phi$ is a
small function of $f$ such that $f\mbox{ and }\phi$ have no common poles. Let
$l\in\mathbb{N}$ and $\psi(z)=f^{(l)}(z)$. If $f(z)\neq 0\mbox{ and
}\psi(z)\neq\phi(z)$ for all $z\in\mathbb{C},$ then $\psi^{\prime}(z)=\phi(z)$
and $\psi^{\prime}(z)=\phi^{\prime}(z)$ have infinitely many solutions.

##  2\.  Proof of Theorem 1.11

Since the proof of Theorem 1.11 is based on Milloux techniques (see [6] p.60),
we need to prove some key lemmas for the proof of Theorem 1.11. Throughout
this paper, we shall denote $f^{(l)}(z)$ by $\psi(z)$, where $l\in\mathbb{N}$.

######  Lemma 2.1.

Let $f\in\mathcal{M}(\mathbb{C})$ and let $\phi$ be a small function of $f.$
Then for $r\to\infty$ outside a set of finite linear measure,

$T\left(r,f\right)\leq\bar{N}\left(r,f\right)+N\left(r,\frac{1}{f}\right)+\bar{N}\left(r,\frac{1}{\psi-\phi}\right)-N^{0}_{2}\left(r,\psi\right)+S(r,f),$
(2.1)

where
$N^{0}_{2}\left(r,\psi\right)=N\left(r,\frac{1}{\psi^{\prime}}\right)-N_{0}\left(r,\frac{1}{\psi^{\prime}}\right)$
and $N_{0}\left(r,\frac{1}{\psi^{\prime}}\right)$ is the counting function of
zeros of $\psi^{\prime}$ which are not zeros of $\psi$.

###### Proof.

By the second fundamental theorem of Nevanlinna for three small functions (see
[6] Theorem. 2.5, also see[4] Theorem. 5.9.1) with $a_{1}=0,a_{2}=\infty\mbox{
and }a_{3}=\phi$, we have

$T(r,f)\leq\bar{N}\left(r,\frac{1}{f}\right)+\bar{N}\left(r,{f}\right)+\bar{N}\left(r,\frac{1}{f-\phi}\right)+S(r,f).$
(2.2)

as $r\to\infty$ outside a set of $r$ of finite linear measure.

Since

$\bar{N}\left(r,\frac{1}{f}\right)={N}\left(r,\frac{1}{f}\right)-{N}\left(r,\frac{1}{f^{\prime}}\right)+{N}_{0}\left(r,\frac{1}{f^{\prime}}\right),$
$~{}~{}~{}~{}\bar{N}\left(r,\frac{1}{f-\phi}\right)={N}\left(r,\frac{1}{f-\phi}\right)-{N}\left(r,\frac{1}{f^{\prime}-\phi^{\prime}}\right)+{N}_{0}\left(r,\frac{1}{f^{\prime}-\phi^{\prime}}\right)$

and

$\bar{N}\left(r,f\right)=N\left(r,f^{\prime}\right)-N\left(r,f\right),$
therefore (2.2) yeilds (after adding
$m\left(r,\frac{1}{f}\right)+m\left(r,\frac{1}{f-\phi}\right)$ to both sides)

$\displaystyle
T\left(r,f\right)+m\left(r,\frac{1}{f}\right)+m\left(r,\frac{1}{f-\phi}\right)$
$\displaystyle\leq
m\left(r,\frac{1}{f}\right)+m\left(r,\frac{1}{f-\phi}\right)$
$\displaystyle+N\left(r,\frac{1}{f}\right)+N\left(r,\frac{1}{f-\phi}\right)+N\left(r,{f^{\prime}}\right)-N\left(r,{f}\right)$
$\displaystyle-N\left(r,\frac{1}{f^{\prime}}\right)+N_{0}\left(r,\frac{1}{f^{\prime}}\right)+N_{0}\left(r,\frac{1}{f^{\prime}-\phi^{\prime}}\right)$
$\displaystyle-N\left(r,\frac{1}{f^{\prime}-\phi^{\prime}}\right)+S(r,f)$

$\displaystyle\Rightarrow
m\left(r,\frac{1}{f}\right)+m\left(r,\frac{1}{f-\phi}\right)+m\left(r,{f}\right)$
$\displaystyle\leq
2T\left(r,f\right)-N_{1}\left(r,f\right)+N_{0}\left(r,\frac{1}{f^{\prime}}\right)$
$\displaystyle+N_{0}\left(r,\frac{1}{f^{\prime}-\phi^{\prime}}\right)-N\left(r,\frac{1}{f^{\prime}-\phi^{\prime}}\right)+S(r,f),$
(2.3)

where
$N_{1}\left(r,f\right)=N\left(r,\frac{1}{f^{\prime}}\right)+2N\left(r,f\right)-N\left(r,f^{\prime}\right).$

Applying (2) to $\psi=f^{(l)}\mbox{ and put }g=\psi-\phi,$ we have

$\displaystyle
m\left(r,\frac{1}{\psi}\right)+m\left(r,\frac{1}{g}\right)+m\left(r,{\psi}\right)$
$\displaystyle\leq
2T\left(r,\psi\right)-N_{1}\left(r,\psi\right)+N_{0}\left(r,\frac{1}{\psi^{\prime}}\right)$
$\displaystyle+N_{0}\left(r,\frac{1}{g^{\prime}}\right)-N\left(r,\frac{1}{g^{\prime}}\right)+S(r,\psi)$
(2.4)

as $r\to\infty$ outside a set of $r$ of finite linear measure.

Since

$N\left(r,\psi^{\prime}\right)-N\left(r,\psi\right)=\bar{N}\left(r,f\right)$

and

$N\left(r,\frac{1}{g}\right)-N\left(r,\frac{1}{g^{\prime}}\right)=\bar{N}\left(r,\frac{1}{g}\right)-N_{0}\left(r,\frac{1}{g^{\prime}}\right)$

and using the first fundamental theorem of Nevanlinna, we have

$\displaystyle 2T\left(r,\psi\right)-N_{1}\left(r,\psi\right)$
$\displaystyle=m\left(r,{\psi}\right)+m\left(r,\frac{1}{g}\right)+N\left(r,\psi\right)+N\left(r,\frac{1}{g}\right)$
$\displaystyle-N\left(r,\frac{1}{\psi^{\prime}}\right)-2N\left(r,\psi\right)+N\left(r,\psi^{\prime}\right)+S(r,f)$
$\displaystyle=m\left(r,{\psi}\right)+m\left(r,\frac{1}{g}\right)+N\left(r,\frac{1}{g}\right)-N\left(r,\frac{1}{\psi^{\prime}}\right)$
$\displaystyle+\bar{N}\left(r,f\right)+S(r,f)$

and hence (2) reduces to

$m\left(r,\frac{1}{\psi}\right)\leq\bar{N}\left(r,\frac{1}{g}\right)-N\left(r,\frac{1}{\psi^{\prime}}\right)+\bar{N}\left(r,f\right)+N_{0}\left(r,\frac{1}{\psi^{\prime}}\right)+S(r,f).$
(2.5)

Also

$\displaystyle T\left(r,f\right)$
$\displaystyle=T\left(r,\frac{1}{f}\right)+O(1)$
$\displaystyle=m\left(r,\frac{\psi}{f\psi}\right)+N\left(r,\frac{1}{f}\right)+O(1)$
$\displaystyle\leq
m\left(r,\frac{\psi}{f}\right)+m\left(r,\frac{1}{\psi}\right)+N\left(r,\frac{1}{f}\right)+O(1)$
$\displaystyle=m\left(r,\frac{1}{\psi}\right)+N\left(r,\frac{1}{f}\right)+S(r,f).$
(2.6)

Now by using (2.5) in (2), we get

$\displaystyle T\left(r,f\right)$
$\displaystyle\leq\bar{N}\left(r,f\right)+N\left(r,\frac{1}{f}\right)+\bar{N}\left(r,\frac{1}{g}\right)-N\left(r,\frac{1}{\psi^{\prime}}\right)+N_{0}\left(r,\frac{1}{\psi^{\prime}}\right)+S(r,f)$
$\displaystyle=\bar{N}\left(r,f\right)+N\left(r,\frac{1}{f}\right)+\bar{N}\left(r,\frac{1}{\psi-\phi}\right)-N^{0}_{2}\left(r,{\psi}\right)+S(r,f)$
(2.7)

as $r\to\infty$ outside a set of $r$ of finite linear measure, where

$N^{0}_{2}\left(r,\psi\right)=N\left(r,\frac{1}{\psi^{\prime}}\right)-N_{0}\left(r,\frac{1}{\psi^{\prime}}\right)=N\left(r,\frac{1}{\psi}\right)-\bar{N}\left(r,\frac{1}{\psi}\right)$

counts only repeated zeros of $\psi$ with multiplicity reduced by $1$. ∎

######  Lemma 2.2.

Let $f\in\mathcal{M}\left(\mathbb{C}\right)$ and let $\phi$ be a small
function of $f$ such that $f\mbox{ and }\phi$ have no common poles. Then

$lN_{1}\left(r,f\right)\leq\bar{N_{2}}\left(r,f\right)+\bar{N}\left(r,\frac{1}{\psi-\phi}\right)+\bar{N}\left(r,\frac{1}{\psi^{\prime}-\phi}\right)+S(r,f),$
(2.8)

where $N_{1}\left(r,f\right)$ is the counting function of simple poles of
$f$and $\bar{N_{2}}\left(r,f\right)$ is the counting function of multiple
poles of $f$ counted once.

The proof of Lemma 2.2 is carried out by following the proof of Lemma 3.1 in
[6] with certain modifications.

Proof of Lemma 2.2. Put

$G(z)={\left\\{\psi^{\prime}(z)-\phi(z)\right\\}^{l+1}\over{\left\\{\phi(z)-\psi(z)\right\\}^{l+2}}}.$

If $z_{0}$ is a simple pole of $f(z),$ then near $z_{0}$ we have

$f(z)={a\over z-z_{0}}+O(1),$

and

$\psi(z)={(-1)^{l}a(l!)\over\left(z-z_{0}\right)^{l+1}}+O(1)$

near $z_{0}$, and hence

$\psi^{\prime}={(-1)^{l+1}a(l+1)!\over\left(z-z_{0}\right)^{l+2}}\left\\{1+O(z-z_{0})^{l+2}\right\\}.$

Since $f$ and $\phi$ have no common poles, therefore near $z_{0},$ we have

$G(z)={(-1)^{l+1}(l+1)^{l+1}\over{al!}}\left\\{1+O(z-z_{0})^{l+1}\right\\}$

which implies that $G(z_{0})\neq 0,\infty,$ and $G^{\prime}(z)$ has a zero of
order atleast $l$ at $z_{0}$ and so

$\displaystyle lN_{1}\left(r,f\right)\leq
N_{0}\left(r,\frac{1}{G^{\prime}}\right).$ (2.9)

Applying Jensen’s formula to ${G^{\prime}/{G}}$ we get

$N\left(r,\frac{G}{G^{\prime}}\right)-N\left(r,\frac{G^{\prime}}{G}\right)=m\left(r,\frac{G^{\prime}}{G}\right)-m\left(r,\frac{G}{G^{\prime}}\right)+O(1).$
(2.10)

Since the only zeros of ${G^{\prime}}/{G}$ are the zeros of $G^{\prime}$ which
are not zeros of $G,$ we have

$\displaystyle
N\left(r,\frac{G}{G^{\prime}}\right)=N_{0}\left(r,\frac{1}{G^{\prime}}\right).$
(2.11)

Also, ${G^{\prime}}/{G}$ has only simple poles at the zeros and poles of $G,$
so

$\displaystyle
N\left(r,\frac{G^{\prime}}{G}\right)=\bar{N}\left(r,G\right)+\bar{N}\left(r,\frac{1}{G}\right).$
(2.12)

Using (2.11) and (2.12) in (2.10), we obtain

$\displaystyle
N_{0}\left(r,\frac{1}{G^{\prime}}\right)-\bar{N}\left(r,\frac{1}{G}\right)-\bar{N}\left(r,{G}\right)=m\left(r,\frac{G^{\prime}}{G}\right)-m\left(r,\frac{G}{G^{\prime}}\right)+O(1).$
(2.13)

Now, from (2.9), (2.10) and (2.13), we have

$\displaystyle lN_{1}\left(r,f\right)$ $\displaystyle\leq
N_{0}\left(r,\frac{1}{G^{\prime}}\right)$
$\displaystyle=\bar{N}\left(r,\frac{1}{G}\right)+\bar{N}\left(r,{G}\right)+m\left(r,\frac{G^{\prime}}{G}\right)-m\left(r,\frac{G}{G^{\prime}}\right)+O(1)$
$\displaystyle\leq\bar{N}\left(r,\frac{1}{G}\right)+\bar{N}\left(r,{G}\right)+m\left(r,\frac{G^{\prime}}{G}\right)+O(1).$
(2.14)

Let $z_{0}$ be a pole of $\psi(z)-\phi(z)$ of order $m$, say. Then near
$z_{0}$

$\psi(z)-\phi(z)={s_{0}(z)\over\left(z-z_{0}\right)^{m}}$

for some function $s_{0}(z)$ analytic in a neighborhood of $z_{0}\mbox{ such
that }s_{0}(z_{0})\neq 0.$

Now, there are two cases:

  * Case 1:

$z_{0}$ is a pole of $f(z)$. Then $m=k+l$, where $k>1$ is the multiplicity of
$z_{0}$ as a pole of $f$. Since $z_{0}$ is not a pole of $\phi,$ we see that
$z_{0}$ is a pole of $\psi^{\prime}(z)-\phi(z)$ of multiplicity $m+1.$
Therefore, near $z_{0}$

$G(z)=t_{0}(z)\left(z-z_{0}\right)^{k-1},$

for some function $t_{0}(z)$ analytic in a neighborhood of $z_{0}\mbox{ such
that }t_{0}(z_{0})\neq 0.$ So $z_{0}$ is a zero of order $k-1$ of $G(z)$.

  * Case 2:

$z_{0}$ is a pole of $\phi(z).$ Then $z_{0}$ is also a pole of
$\psi^{\prime}(z)-\phi(z)$ of multiplicity $m.$ Therefore, near $z_{0}$

$G(z)=t_{1}(z)\left(z-z_{0}\right)^{m},$

for some function $t_{1}(z)$ analytic in a neighborhood of $z_{0}\mbox{ such
that }t_{1}(z_{0})\neq 0.$ This shows that $z_{0}$ is a pole of $G(z)$ of the
same multiplicity as that of $\phi(z).$

Similarly, looking at the poles of $\psi^{\prime}(z)-\phi(z),$ we obtain the
same conclusion as in the case of poles of $\psi(z)-\phi(z).$

Next, correspnding to the zeros of $\psi(z)-\phi(z)$ and
$\psi^{\prime}(z)-\phi(z),$ we have the following three cases:

  * Case 1:

$z_{0}$ is a zero of $\psi(z)-\phi(z)$ but it is not a zero of
$\psi^{\prime}(z)-\phi(z).$ Then $z_{0}$ is a pole of $G(z)$.

  * Case 2:

$z_{0}$ is zero of $\psi^{\prime}(z)-\phi(z)$ but it is not a zero of
$\psi(z)-\phi(z).$ Then $z_{0}$ is a zero of $G(z)$.

  * Case 3:

$z_{0}$ is a common zero of $\psi^{\prime}(z)-\phi(z)\mbox{ and
}\psi(z)-\phi(z)$. Let $j$ and $k$ be the multiplicities of $z_{0}$ as a zero
of $\psi^{\prime}(z)-\phi(z)\mbox{ and }\psi(z)-\phi(z)$, respectively. Then
near $z_{0}$,

$G(z)=t_{2}(z)\left(z-z_{0}\right)^{(l+1)j-(l+2)k}$

for some function $t_{2}(z)$ analytic in a neighborhood of $z_{0}\mbox{ such
that }t_{2}(z_{0})\neq 0$.

Thus $z_{0}$ is a pole of $G(z)$ if $k>\frac{l+1}{l+2}j$ and $z_{0}$ is a zero
of $G(z)$ if $k<\frac{l+1}{l+2}j$.

Let $N(r,\frac{1}{f},\frac{1}{g^{0}})$ be the counting function of zeros of
$f$ which are not zeros of $g$ , $N(r,\frac{1}{f},\frac{1}{g})$ be the
counting function corresponding to the common zeros of $f\mbox{ and }g$ and
$N^{(\alpha)}(r,\frac{1}{f},\frac{1}{g})$ be the counting function
corresponding to the common zeros of $f$ and $g$, such that the
$m(f,z_{0})>\alpha m(g,z_{0})$, where by $m(f,z_{0})$ we denote the
multiplicity of $z_{0}$ as a zero of $f$. With these notations and the
preceding arguments , we find that

$\displaystyle\bar{N}\left(r,\frac{1}{G}\right)$
$\displaystyle\leq\bar{N}_{2}(r,f)+\bar{N}(r,\phi)+\bar{N}\left(r,\frac{1}{\psi^{\prime}-\phi},\frac{1}{(\psi-\phi)^{0}}\right)$
$\displaystyle+\bar{N}^{\left(\frac{l+2}{l+1}\right)}\left(r,\frac{1}{\psi^{\prime}-\phi},\frac{1}{\psi-\phi}\right)$
(2.15)

and

$\displaystyle\bar{N}(r,G)$
$\displaystyle\leq\bar{N}\left(r,\frac{1}{\psi-\phi},\frac{1}{(\psi^{\prime}-\phi)^{0}}\right)+\bar{N}^{\left(\frac{l+1}{l+2}\right)}\left(r,\frac{1}{\psi-\phi},\frac{1}{\psi^{\prime}-\phi}\right).$
(2.16)

Note that

$\displaystyle\bar{N}\left(r,\frac{1}{\psi-\phi},\frac{1}{(\psi^{\prime}-\phi)^{0}}\right)+\bar{N}^{\left(\frac{l+1}{l+2}\right)}\left(r,\frac{1}{\psi-\phi},\frac{1}{\psi^{\prime}-\phi}\right)$
$\displaystyle+\bar{N}\left(r,\frac{1}{\psi^{\prime}-\phi},\frac{1}{(\psi-\phi)^{0}}\right)+\bar{N}^{\left(\frac{l+2}{l+1}\right)}\left(r,\frac{1}{\psi^{\prime}-\phi},\frac{1}{\psi-\phi}\right)$
$\displaystyle\leq\bar{N}\left(r,\frac{1}{\psi-\phi}\right)+\bar{N}\left(r,\frac{1}{\psi^{\prime}-\phi}\right).$

Therefore, using (2) and (2.16) in (2), we get

$\displaystyle lN_{1}\left(r,f\right)$
$\displaystyle\leq\bar{N}_{2}(r,f)+\bar{N}(r,\phi)+\bar{N}\left(r,\frac{1}{\psi-\phi}\right)+\bar{N}\left(r,\frac{1}{\psi^{\prime}-\phi}\right)+m\left(r,\frac{G^{\prime}}{G}\right)+O(1).$
(2.17)

Since $T(r,\phi)=S(r,f)$ and $S(r,\psi)=S(r,f)$, by Theorem 3.1 in [6], we
have

$m\left(r,\frac{G^{\prime}}{G}\right)=S(r,f).$

Thus from (2.17), it follows that

$\displaystyle lN_{1}\left(r,f\right)$
$\displaystyle\leq\bar{N}_{2}(r,f)+\bar{N}\left(r,\frac{1}{\psi-\phi}\right)+\bar{N}\left(r,\frac{1}{\psi^{\prime}-\phi}\right)+S(r,f).$
(2.18)

∎

######  Lemma 2.3.

Let $f\in\mathcal{M}\left(\mathbb{C}\right)$ and let $\phi$ be a small
function of $f$ such that $f\mbox{ and }\phi$ have no common poles. Then

$lN_{1}\left(r,f\right)\leq\bar{N_{2}}\left(r,f\right)+\bar{N}\left(r,\frac{1}{\psi-\phi}\right)+N_{0}\left(r,\frac{1}{\psi^{\prime}-\phi^{\prime}}\right)+S(r,f),$
(2.19)

where $N_{1}\left(r,f\right)$ is the counting function of simple poles of $f$,
$\bar{N_{2}}\left(r,f\right)$ is the counting function of multiple poles of
$f$ counted once and
$N_{0}\left(r,\frac{1}{\psi^{\prime}-\phi^{\prime}}\right)$ is the counting
function of zeros of $\psi^{\prime}-\phi^{\prime}$ which are not repeated
zeros of $\psi-\phi$.

###### Proof.

Define

$G(z)={\left\\{\psi^{\prime}(z)-\phi^{\prime}(z)\right\\}^{l+1}\over{\left\\{\phi(z)-\psi(z)\right\\}^{l+2}}}.$

Then as in the proof of Lemma 2.2 above, we again arrive at (2).

Next, to find the distribution of poles and zeros of $G(z)$, we proceed as
follows;

Put,

$h(z)=\psi(z)-\phi(z).$

If $z_{0}$ is a pole of $h(z)$ of order $m,$ then near $z_{0},$

$h(z)={s(z)\over\left(z-z_{0}\right)^{m}}\mbox{ and
}h^{\prime}(z)={t(z)\over\left(z-z_{0}\right)^{m+1}}~{},$

where $s(z)\mbox{ and }t(z)$ are functions analytic in a neighborhood of
$z_{0}$ and both have no zeros at $z_{0}.$ So,

$\displaystyle G(z)={w(z)\over\left(z-z_{0}\right)^{l+1-m}}$ (2.20)

for some function $w(z)$ analytic in a neighborhood of $z_{0}\mbox{ such that
}w(z_{0})\neq 0.$

Next if $z_{0}$ is a zero of $h(z),$ then near $z_{0}$,
$h(z)={l(z)\left(z-z_{0}\right)^{m}}$ and so

$\displaystyle G(z)={m(z)\over\left(z-z_{0}\right)^{l+1+m}}$ (2.21)

where $l(z)\mbox{ and }m(z)$ are functions analytic in a neighborhood of
$z_{0}$ and both have no zeros at $z_{0}.$

From (2.21) and (2.20) we see that the only poles of $G(z)$ occur at

  * (i)

the roots of $h(z)=0$ and

  * (ii)

the poles of $\phi(z)$ of multiplicity less than $l+1$.

Therefore,

$\displaystyle\bar{N}\left(r,G\right)\leq\bar{N}\left(r,\frac{1}{\psi-\phi}\right)+\bar{N}\left(r,\phi\right)-\bar{N}_{l+1}\left(r,\phi\right),$
(2.22)

where $\bar{N}_{k}\left(r,\phi\right)$ is the counting function of poles of
$\phi(z)$ which have multiplicity atleast $k$, each pole is counted once.

Since the zeros of $G(z)$ occur at

  * (i)

the roots of $h^{\prime}(z)=0$ which are not the roots of $h(z)=0$

  * (ii)

multiple poles of $f(z)$ and

  * (iii)

poles of $\phi$ of multiplicity greater than $l+1$,

therefore,

$\displaystyle\bar{N}\left(r,\frac{1}{G}\right)\leq\bar{N}_{0}\left(r,\frac{1}{\psi^{\prime}-\phi^{\prime}}\right)+\bar{N}_{2}\left(r,f\right)+\bar{N}_{l+1}\left(r,\phi\right).$
(2.23)

Adding (2.22) and (2.23), we have;

$\displaystyle\bar{N}\left(r,G\right)+\bar{N}\left(r,\frac{1}{G}\right)\leq\bar{N_{2}}\left(r,f\right)+\bar{N}\left(r,\phi\right)+\bar{N}\left(r,\frac{1}{\psi-\phi}\right)+N_{0}\left(r,\frac{1}{\psi^{\prime}-\phi^{\prime}}\right).$
(2.24)

Since $T(r,\phi)=S(r,f)$ and $S(r,\psi)=S(r,f)$, by Theorem $3.1$ in [6], we
have

$m\left(r,\frac{G^{\prime}}{G}\right)=S(r,f).$

Thus, from (2) and (2.24), it follows that

$lN_{1}\left(r,f\right)\leq\bar{N_{2}}\left(r,f\right)+\bar{N}\left(r,\frac{1}{\psi-\phi}\right)+N_{0}\left(r,\frac{1}{\psi^{\prime}-\phi^{\prime}}\right)+S(r,f).$
(2.25)

∎

######  Lemma 2.4.

Let $f,\psi\mbox{ and }\phi$ be as in Lemma 2.2. Then

$\displaystyle(a)~{}T\left(r,f\right)$
$\displaystyle\leq\left(2+\frac{1}{l}\right)N\left(r,\frac{1}{f}\right)+\left(2+\frac{2}{l}\right)\bar{N}\left(r,\frac{1}{\psi-\phi}\right)$
$\displaystyle+\frac{1}{l}\bar{N}\left(r,\frac{1}{\psi^{\prime}-\phi}\right)-\left(2+\frac{1}{l}\right)N^{0}_{2}\left(r,\psi\right)+S(r,f).$
$\displaystyle(b)~{}T\left(r,f\right)$
$\displaystyle\leq\left(2+\frac{1}{l}\right)N\left(r,\frac{1}{f}\right)+\left(2+\frac{2}{l}\right)\bar{N}\left(r,\frac{1}{\psi-\phi}\right)$
$\displaystyle+\frac{1}{l}N_{0}\left(r,\frac{1}{\psi^{\prime}-\phi^{\prime}}\right)-\left(2+\frac{1}{l}\right)N^{0}_{2}\left(r,\psi\right)+S(r,f).$

###### Proof.

By Lemma 2.1, we have

$\displaystyle N_{1}\left(r,f\right)+2\bar{N_{2}}\left(r,f\right)$
$\displaystyle\leq N\left(r,f\right)$ $\displaystyle\leq T\left(r,f\right)$
$\displaystyle\leq\bar{N}\left(r,f\right)+N\left(r,\frac{1}{f}\right)+\bar{N}\left(r,\frac{1}{\psi-\phi}\right)-N^{0}_{2}\left(r,\psi\right)+S(r,f).$
(2.26)

Since
$\bar{N}\left(r,f\right)=N_{1}\left(r,f\right)+\bar{N_{2}}\left(r,f\right),$
from (2) we have

$\displaystyle\bar{N_{2}}\left(r,f\right)\leq
N\left(r,\frac{1}{f}\right)+\bar{N}\left(r,\frac{1}{\psi-\phi}\right)-N^{0}_{2}\left(r,\psi\right)+S(r,f).$
(2.27)

Using (2.27) in Lemma 2.2, we obtain

$\displaystyle N_{1}\left(r,f\right)$
$\displaystyle\leq\frac{1}{l}N\left(r,\frac{1}{f}\right)+\frac{2}{l}\bar{N}\left(r,\frac{1}{\psi-\phi}\right)-\frac{1}{l}N^{0}_{2}\left(r,\psi\right)$
$\displaystyle+\frac{1}{l}\bar{N}\left(r,\frac{1}{\psi^{\prime}-\phi}\right)+S(r,f).$
(2.28)

Now from (2.27) and (2) it follows that

$\displaystyle\bar{N}\left(r,f\right)$
$\displaystyle=N_{1}\left(r,f\right)+\bar{N_{2}}\left(r,f\right)$
$\displaystyle\leq
N_{1}\left(r,f\right)+N\left(r,\frac{1}{f}\right)+\bar{N}\left(r,\frac{1}{\psi-\phi}\right)-N^{0}_{2}\left(r,\psi\right)+S(r,f)$
$\displaystyle\leq\left(1+\frac{1}{l}\right)N\left(r,\frac{1}{f}\right)+\left(1+\frac{2}{l}\right)\bar{N}\left(r,\frac{1}{\psi-\phi}\right)-\left(1+\frac{1}{l}\right)N^{0}_{2}\left(r,\psi\right)$
$\displaystyle+\frac{1}{l}\bar{N}\left(r,\phi\right)+\frac{1}{l}\bar{N}\left(r,\frac{1}{\psi^{\prime}-\phi}\right)+S(r,f).$
(2.29)

Now in view of (2), Lemma 2.1 yields

$\displaystyle T\left(r,f\right)$
$\displaystyle\leq\left(2+\frac{1}{l}\right)N\left(r,\frac{1}{f}\right)+\left(2+\frac{2}{l}\right)\bar{N}\left(r,\frac{1}{\psi-\phi}\right)$
$\displaystyle+\frac{1}{l}\bar{N}\left(r,\frac{1}{\psi^{\prime}-\phi}\right)-\left(2+\frac{1}{l}\right)N^{0}_{2}\left(r,\psi\right)+S(r,f),$

which proves $(a)$.

The conclusion $(b)$ follows by using Lemma 2.3 instead of Lemma 2.2 in the
proof of $(a)$, above.

∎

Proof of Theorem 1.11:  Since $N^{0}_{2}\left(r,\psi\right)\geq 0$, by Lemma
2.4$(a)$ we have

$\displaystyle T\left(r,f\right)$ $\displaystyle\leq
3N\left(r,\frac{1}{f}\right)+4\bar{N}\left(r,\frac{1}{\psi-\phi}\right)+\bar{N}\left(r,\frac{1}{\psi^{\prime}-\phi}\right)+S(r,f).$
(2.30)

Since $f\mbox{ and }\psi-\phi$ have no zeros,
$N\left(r,\frac{1}{f}\right)=0\mbox{ and
}\bar{N}\left(r,\frac{1}{\psi-\phi}\right)=0.$ Therefore, (2.30) reduces to

$\displaystyle
T\left(r,f\right)\leq\bar{N}\left(r,\frac{1}{\psi^{\prime}-\phi}\right)+S(r,f).$
(2.31)

Since $f\in\mathcal{M}\left(\mathbb{C}\right)$ is transcendental, (2.31)
implies that $\psi^{\prime}(z)=\phi(z)$ has infinitely many solutions.
Similarly, Lemma 2.4$(b)$ implies that $\psi^{\prime}(z)=\phi^{\prime}(z)$ has
infinitely many solutions. ∎

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