## 1 Introduction

Let Γ be the subset of the upper half unit sphere. The set $${\mathbf {R}}_{+}\times\Gamma$$ in $${\mathbf {R}}^{n}$$ is called a cone. We denote it by $$\mathfrak{C}_{n}(\Gamma)$$, where $$\Gamma\subset{\mathbf {S}}_{1}$$. The sets $$I\times\Gamma$$ and $$I\times \partial{\Gamma}$$ with an interval on R are denoted by $$\mathfrak{C}_{n}(\Gamma;I)$$ and $$\mathfrak{S}_{n}(\Gamma;I)$$, respectively. We denote $$\mathfrak {C}_{n}(\Gamma)\cap S_{R}$$ and $$\mathfrak{S}_{n}(\Gamma; (0,+\infty))$$ by $$\mathfrak {S}_{n}(\Gamma; R)$$ and $$\mathfrak{S}_{n}(\Gamma)$$, respectively.

Furthermore, we denote by (resp. $$dS_{R}$$) the $$(n-1)$$-dimensional volume elements induced by the Euclidean metric on $$\partial{\mathfrak{C}_{n}(\Gamma)}$$ (resp. $$S_{R}$$) and by dw the elements of the Euclidean volume in $${\mathbf {R}}^{n}$$.

It is well known (see, e.g., [2], p.41) that

\begin{aligned} &{\Delta^{*}\varphi(\Theta)+\lambda\varphi(\Theta)=0\quad \textrm{in } \Gamma,} \\ \\ &{\varphi(\Theta)=0\quad\textrm{on } \partial{\Gamma},} \end{aligned}
(1.1)

where $$\Delta^{*}$$ is the Laplace-Beltrami operator. We denote the least positive eigenvalue of this boundary value problem (1.1) by λ and the normalized positive eigenfunction corresponding to λ by $$\varphi(\Theta)$$, $$\int_{\Gamma}\varphi^{2}(\Theta)\,dS_{1}=1$$.

We remark that the function $$r^{\aleph^{\pm}}\varphi(\Theta)$$ is harmonic in $$\mathfrak{C}_{n}(\Gamma)$$, belongs to the class $$C^{2}(\mathfrak{C}_{n}(\Gamma )\backslash\{O\})$$ and vanishes on $$\mathfrak{S}_{n}(\Gamma)$$, where

$$2\aleph^{\pm}=-n+2\pm\sqrt{(n-2)^{2}+4\lambda}.$$

For simplicity we shall write χ instead of $$\aleph^{+}-\aleph^{-}$$.

For simplicity we shall assume that the boundary of the domain Γ is twice continuously differentiable, $$\varphi\in C^{2}(\overline{\Gamma})$$ and $$\frac{\partial\varphi}{\partial n}>0$$ on Γ. Then (see [3], p.7-8)

$$\operatorname{dist}(\Theta,\partial{\Gamma})\approx\varphi(\Theta),$$
(1.2)

where $$\Theta\in\Gamma$$.

Let $$\delta(P)=\operatorname{dist}(P,\partial{\mathfrak{C}_{n}(\Gamma)})$$, we have

$$\varphi(\Theta)\approx\delta(P),$$
(1.3)

for any $$P=(1,\Theta)\in\Gamma$$ (see [4]).

Let $$u(r,\Theta)$$ be a function on $$\mathfrak{C}_{n}(\Gamma)$$. For any given $$r\in{\mathbf {R}}_{+}$$, the integral

$$\int_{\Gamma}u(r,\Theta)\varphi(\Theta)\,d S_{1},$$

is denoted by $$\mathcal{N}_{u}(r)$$, when it exists. The finite or infinite limits

$$\lim_{r\rightarrow\infty}r^{-\aleph^{+}}\mathcal{N}_{u}(r)\quad\textrm{and}\quad \lim_{r\rightarrow0}r^{-\aleph^{-}}\mathcal{N}_{u}(r)$$

are denoted by $$\mathscr{U}_{u}$$ and $$\mathscr{V}_{u}$$, respectively, when they exist.

### Remark 1

A function $$g(t)$$ on $$(0,\infty)$$ is $$\mathbb{A}_{d_{1},d_{2}}$$-convex if and only if $$g(t)t^{d_{2}}$$ is a convex function of $$t^{d}$$ $$(d=d_{1}+d_{2})$$ on $$(0,\infty)$$, or, equivalently, if and only if $$g(t)t^{-d_{1}}$$ is a convex function of $$t^{-d}$$ on $$(0,\infty)$$.

### Remark 2

$$\mathcal{N}_{u}(r)$$ is a $$\mathbb{A}_{\aleph^{+},\gamma-1}$$-convex on $$(0,\infty)$$, where u is a subharmonic function on $$\mathfrak{C}_{n}(\Gamma)$$ such that

$$\limsup_{P\in\mathfrak{C}_{n}(\Gamma),P\rightarrow Q \in \partial{\mathfrak{C}_{n}(\Gamma)}}u(P)\leq c,$$
(1.4)

where c is a nonnegative number (see [5]).

The function

$$\mathbb{P}_{\mathfrak{C}_{n}(\Gamma)}(P,Q)=\frac{\partial\mathbb {G}_{\mathfrak{C}_{n}(\Gamma)}(P,Q)}{\partial n_{Q}}$$

is called the ordinary Poisson kernel, where $$\mathbb{G}_{\mathfrak{C}_{n}(\Gamma)}$$ is the Green function.

The Poisson integral of g relative to $$\mathfrak{C}_{n}(\Gamma)$$ is defined by

$$\mathbb{PI}_{\mathfrak{C}_{n}(\Gamma)} [g](P)=\frac{1}{c_{n}} \int_{\mathfrak{S}_{n}(\Gamma)}\mathbb {P}_{\mathfrak{C}_{n}(\Gamma)}(P,Q)g(Q)\,d\sigma,$$

where g is a continuous function on $$\partial{\mathfrak{C}_{n}(\Gamma)}$$ and $$\frac{\partial}{\partial n_{Q}}$$ denotes the differentiation at Q along the inward normal into $$\mathfrak{C}_{n}(\Gamma)$$.

We set functions f satisfying

$$\int_{\mathfrak{S}_{n}(\Gamma)}\frac{ \vert f(t,\Phi) \vert ^{p}}{1+t^{\gamma }}\,d\sigma< \infty,$$
(1.5)

where $$-1< p<+\infty$$ and

$$\frac{-\aleph^{+}-n+2}{p}< \gamma< \frac{-\aleph^{+}-n+2}{p}+n-1.$$

Let $$-1< p<+\infty$$. we denote $$\mathcal{A}_{\Gamma}$$ the class of all measurable functions $$g(t,\Phi)$$ $$(Q=(t,\Phi)=(Y, y_{n})\in \mathfrak{C}_{n}(\Gamma))$$ satisfying the following inequality:

$$\int_{\mathfrak{C}_{n}(\Gamma)}\frac{ \vert g(t,\Phi) \vert ^{p-1}\varphi }{1+t^{\gamma-3}}\,dw< \infty$$

and the class $$\mathcal{B}_{\Gamma}$$, consists of all measurable functions $$h(t,\Phi)$$ $$((t,\Phi)=(Y, y_{n})\in\mathfrak{S}_{n}(\Gamma))$$ satisfying

$$\int_{\mathfrak{S}_{n}(\Gamma)}\frac{ \vert h(t,\Phi) \vert ^{q}}{1+t^{\gamma}}\frac{\partial\varphi}{\partial n}\,d\sigma < \infty,$$

where $$q>0$$.

We will also consider the class of all continuous functions $$u(t,\Phi)$$ $$((t,\Phi)\in\overline{\mathfrak{C}_{n}(\Gamma)})$$ harmonic in $$\mathfrak{C}_{n}(\Gamma)$$ with $$u^{+}(t,\Phi)\in \mathcal{A}_{\Gamma}$$ $$((t,\Phi)\in\mathfrak{C}_{n}(\Gamma))$$ and $$u^{+}(t,\Phi)\in\mathcal{B}_{\Gamma}$$ $$((t,\Phi)\in\mathfrak {S}_{n}(\Gamma))$$ is denoted by $$\mathcal{C}_{\Gamma}$$ (see [6]).

In 2015, Jiang, Hou and Peixoto-de-Büyükkurt (see [7]) obtained the following result.

### Theorem A

Let g be a measurable function on $$\partial{T_{n}}$$ such that

$$\int_{\partial{T_{n}}}\bigl(1+ \vert Q \vert \bigr)^{2-n} \bigl\vert g(Q) \bigr\vert \,dQ< \infty.$$

Then the harmonic function $$\mathbb{PI}_{T_{n}}[g]$$ satisfies $$\mathbb{PI}_{T_{n}}[g](P)=o(r^{2}\sec^{n-3}\theta_{1})$$ as $$r\rightarrow\infty$$ in $$T_{n}$$.

Recently, Wang, Huang and N. Yamini (see [8]) generalized Theorem  A to the conical case.

### Theorem B

Let g be a continuous function on $$\partial {\mathfrak{C}_{n}(\Gamma)}$$ satisfying (1.5) with $$p=q=1$$ and $$\gamma=\aleph^{+}+1-\aleph^{-}$$. Then

$$\mathscr{U}_{\mathbb{PI}_{\mathfrak{C}_{n}(\Gamma)}[g]}=\mathscr {U}_{\mathbb{PI}_{\mathfrak{C}_{n}(\Gamma)}[ \vert g \vert ]}=0.$$

The remainder of the paper is organized as follows: in Section  2, we shall give our main theorem; in Section  3, some necessary lemmas are given; in Section  4, we shall prove the main result.

## 2 Main result

In this section, we give the main result of this paper.

Our main aim is to give a least harmonic majorant of a nonnegative subharmonic function on $$\mathfrak{C}_{n}(\Gamma)$$.

### Theorem 1

Let u be a function subharmonic in $$\mathfrak {C}_{n}(\Gamma)$$ and $$u'$$ be the restriction of u to $$\partial{\mathfrak{C}_{n}(\Gamma)}$$. If $$u'$$ satisfy (1.5) and $$-1\leq\mathscr{U}_{u}\leq1$$ then

$$u(P)\leq h_{u}(P)$$
(2.1)

for any $$P=(r,\Theta)\in\mathfrak{C}_{n}(\Gamma)$$, where $$h_{u}(P)$$ is the least harmonic majorant of u on $$\mathfrak{C}_{n}(\Gamma)$$ and has the following expression:

$$h_{u}(P)=\mathbb{PI}_{\mathfrak{C}_{n}(\Gamma)}\bigl[u'\bigr](P)+ \mathscr {V}_{u}r^{\aleph^{-}}\varphi(\Theta)+\mathscr{U}_{u}r^{\aleph ^{+}} \varphi(\Theta).$$

### Remark 3

Theorem 1 solves a theoretical question raised in connection with the application of Dirichlet-Sch type inequality, obtained by Huang (see [1]), which has been already applied to obtain multiplicity results for boundary value problems in several recent papers.

## 3 Main lemmas

In order to prove our main result, we need the following lemmas.

### Lemma 1

see [1]

Let u be a function subharmonic on $$\mathfrak{C}_{n}(\Gamma)$$ satisfying (1.4). Then the limit $$\mathscr{U}_{u}$$ $$(-1<\mathscr{U}_{u}\leq1)$$ exists.

### Lemma 2

Let u be a function subharmonic on $$\mathfrak{C}_{n}(\Gamma)$$ satisfying (1.4) and

$$\mathscr{U}_{u^{+}}\leq1\quad\textrm{and}\quad \mathscr {U}_{u^{+}}< +\infty.$$
(3.1)

Then

$$u(r,\Theta)\leq\mathscr{V}_{u^{+}}r^{\aleph^{-}}\varphi( \Theta)+ \mathscr{U}_{u^{+}}r^{\aleph^{+}}\varphi(\Theta).$$
(3.2)

### Proof

Take any $$(r,\Theta)\in\mathfrak{C}_{n}(\Gamma)$$ and any pair of numbers $$\tau_{1}$$, $$\tau_{2}$$ $$(0<\tau_{1}<r<\tau_{2}<+\infty)$$. We define a boundary function on $$\partial{\mathfrak{C}_{n}(\Gamma;(\tau_{1},\tau_{2}))}$$ by

$$\nu(r,\Theta)= \textstyle\begin{cases} u(\tau_{i},\Theta) & \mbox{on } \{\tau_{i}\}\times\Gamma\ (i=1,2),\\ 0 & \mbox{on } [\tau_{1},\tau_{2}]\times\partial{\Gamma}. \end{cases}$$

If we denote Schrödinger PWB solution of the Dirichlet-Sch problem on $$\mathfrak{C}_{n}(\Gamma;(\tau_{1},\tau_{2}))$$ with ν by $$H_{\nu}((r,\Theta);\mathfrak{C}_{n}(\Gamma;(\tau_{1},\tau_{2})))$$, then we have

\begin{aligned} u(r,\Theta) \leq& H_{\nu}\bigl((r,\Theta); \mathfrak{C}_{n}\bigl(\Gamma;(\tau_{1},\tau_{2}) \bigr)\bigr) \\ \leq& \int_{\Gamma}u^{+}(\tau_{1},\Theta) \frac{\partial \mathbb{G}_{\mathfrak{C}_{n}(\Gamma;(\tau_{1},\tau_{2}))}((\tau_{1},\Phi ),(r,\Theta))}{\partial R}\tau_{1}^{n-1}\,dS_{1} \\ &{} - \int_{\Gamma}u^{+}(\tau_{2},\Theta) \frac{\partial \mathbb{G}_{\mathfrak{C}_{n}(\Gamma;(\tau_{1},\tau_{2}))}((\tau_{2},\Phi ),(r,\Theta))}{\partial R}\tau_{2}^{n-1}\,dS_{1}, \end{aligned}

which shows that (3.2) holds from (3.1). □

### Lemma 3

Let g be a locally integrable function on $$\partial{\mathfrak{C}_{n}(\Gamma)}$$ satisfying (1.5) and u be a subharmonic function on $$\mathfrak{C}_{n}(\Gamma)$$ satisfying

$$-1\leq\liminf_{P\in\mathfrak {C}_{n}(\Gamma), P\rightarrow Q\in\partial{\mathfrak{C}_{n}(\Gamma)}}\bigl\{ u(P)- \mathbb{PI}_{\mathfrak {C}_{n}(\Gamma)}[g](P)\bigr\} \leq1$$
(3.3)

and

$$\liminf_{P\in\mathfrak{C}_{n}(\Gamma), P\rightarrow Q\in\partial{\mathfrak{C}_{n}(\Gamma)}}\bigl\{ u^{+}(P)-\mathbb {PI}_{\mathfrak{C}_{n}(\Gamma)}\bigl[ \vert g \vert \bigr](P)\bigr\} \leq0.$$
(3.4)

Then the limits $$\mathscr{U}_{u}$$ and $$\mathscr{V}_{u^{+}}$$ ($$-\infty<\mathscr{U}_{u}\leq1$$, $$0\leq\mathscr{U}_{u^{+}}\leq+\infty$$) exist, and if (3.1) is satisfied, then

$$u(P)\leq \mathbb{PI}_{\mathfrak{C}_{n}(\Gamma)}[g](P)+\mathscr {V}_{u^{+}}r^{\aleph^{-}}\varphi(\Theta)+\mathscr {U}_{u^{+}}r^{\aleph^{+}} \varphi(\Theta)$$
(3.5)

for any $$P=(r,\Theta)\in\mathfrak{C}_{n}(\Gamma)$$.

### Proof

Put

$$U(P)=u(P)-\mathbb{PI}_{\mathfrak{C}_{n}(\Gamma)}[g](P)\quad\textrm {and}\quad U'(P)=u^{+}(P)- \mathbb{PI}_{\mathfrak{C}_{n}(\Gamma)}\bigl[ \vert g \vert \bigr](P)$$

on $$\mathfrak{C}_{n}(\Gamma)$$. From (3.3) and (3.4) we have

$$\liminf_{P\in\mathfrak{C}_{n}(\Gamma), P\rightarrow Q}U(P)\leq-1\quad \textrm{and}\quad \liminf _{P\in\mathfrak {C}_{n}(\Gamma), P\rightarrow Q}U'(P)\leq-1.$$

Hence it follows from Lemma 1 that the limits $$\mathscr{U}_{U}$$ and $$\mathscr{V}_{U'}$$ ($$-1<\mathscr{U}_{U}\leq1$$, $$0\leq\mathscr{V}_{U'}\leq1$$) exist. So Theorem B gives the existence of the limits $$\mathscr{U}_{u}$$, $$\mathscr{V}_{u^{+}}$$,

$$\mathscr{U}_{U}=\mathscr{V}_{u}\quad\textrm{and}\quad \mathscr {U}_{U'}=\mathscr{V}_{u^{+}}.$$
(3.6)

Since $$0\leq U^{+}(P)\leq u^{+}(P)+(\mathbb{PI}_{\mathfrak {C}_{n}(\Gamma)}[g])^{-}(P)$$ on $$\mathfrak{C}_{n}(\Gamma)$$, it also follows from Theorem B and (3.1) that

$$\mathscr{V}_{U^{+}}\leq\mathscr{V}_{u^{+}}< \infty,$$

which together with Lemma 2 gives the conclusion. □

### Lemma 4

Let g be a lower semi-continuous function on $$\partial{\mathfrak{C}_{n}(\Gamma)}$$ satisfying (1.5) and u be a superharmonic function on $$\mathfrak{C}_{n}(\Gamma)$$ such that

$$\liminf_{P\in\mathfrak{C}_{n}(\Gamma), P\rightarrow Q}u(P)\leq g(Q)+c$$
(3.7)

for any $$Q\in\partial{\mathfrak{C}_{n}(\Gamma)}$$ and c is a positive number. Then the limit $$\mathscr{U}_{u}$$ ($$-1\leq\mathscr{U}_{u}\leq+1$$) exists, and if $$\mathscr{U}_{u}<+\infty$$, then

$$u(P)\leq \mathbb{PI}_{\mathfrak{C}_{n}(\Gamma)}[g](P)+\mathscr{V}_{u}r^{\aleph ^{-}} \varphi(\Theta)+\mathscr{U}_{u}r^{\aleph^{+}}\varphi(\Theta)$$

for any $$P=(r,\Theta)\in\mathfrak{C}_{n}(\Gamma)$$.

### Proof

Since −g is upper semi-continuous function in $$\partial{\mathfrak{C}_{n}(\Gamma)}$$, it follows from [8], p.3, that

$$\liminf_{P\in\mathfrak{C}_{n}(\Gamma), P\rightarrow Q}\mathbb{PI}_{\mathfrak{C}_{n}(\Gamma)}[g](P) \geq g(Q)-c.$$
(3.8)

We see from (3.7) and (3.8) that

$$-1\leq\limsup_{P\in\mathfrak{C}_{n}(\Gamma), P\rightarrow Q}\bigl\{ u(P)-\mathbb{PI}_{\mathfrak{C}_{n}(\Gamma)}[g](P) \bigr\} \leq1,$$

which gives (3.3). Since g and u are positive, (3.4) also holds. Lemma 4 is proved. □

### Lemma 5

Let u be a subharmonic function in $$\overline {\mathfrak{C}_{n}(\Gamma)}$$ such that $$u'=u|\partial{\mathfrak{C}_{n}(\Gamma)}$$ satisfies (1.5). Then $$\mathbb{PI}_{\mathfrak{C}_{n}(\Gamma)}[u'](P)\leq h(P)$$ on $$\mathfrak{C}_{n}(\Gamma)$$, where $$h(P)$$ is the any harmonic majorant of u on  $$\mathfrak{C}_{n}(\Gamma)$$.

### Proof

Take any $$P'=(r',\Theta')\in \mathfrak{C}_{n}(\Gamma)$$. Let ϵ be any positive number. In the same way as in the proof of Lemma 2, we can choose R such that

$$\int_{\mathfrak{S}_{n}(\Gamma;(R,\infty))}\mathbb{P}_{\mathfrak {C}_{n}(\Gamma)}\bigl(P',Q \bigr)u'(Q)\,d\sigma< \frac{\epsilon}{2}.$$
(3.9)

Further, take an integer j $$(j>R)$$ such that (see [7])

$$\int_{\mathfrak{S}_{n}(\Gamma;(0,R))}\frac{\partial\Gamma_{j}(P',Q) }{\partial n_{Q}}u'(Q)\,d\sigma< \frac{\epsilon}{2}.$$
(3.10)

Since

$$\int_{\mathfrak{S}_{n}(\Gamma;(0,R))}\frac{\partial \mathbb{G}_{\mathfrak{C}_{n}(\Gamma;(0,j))}(P,Q) }{\partial n_{Q}}u'(Q)\,d\sigma\leq H_{u}\bigl(P;\mathfrak{C}_{n}\bigl(\Gamma;(0,j)\bigr)\bigr)$$

for any $$P\in\mathfrak{C}_{n}(\Gamma;(0,j))$$, we have from (3.9) and (3.10)

\begin{aligned} &{\mathbb{PI}_{\mathfrak{C}_{n}(\Gamma)}\bigl[u'\bigr] \bigl(P'\bigr)-H_{u}\bigl(P';\mathfrak {C}_{n}\bigl(\Gamma;(0,j)\bigr)\bigr)} \\ &{\quad\leq \int_{\mathfrak{S}_{n}(\Gamma;(0,R))}\frac{\partial \Gamma_{j}(P',Q) }{\partial n_{Q}}u'(Q)\,d\sigma} \\ &{\qquad {}+\frac{1}{c_{n}} \int_{\mathfrak{S}_{n}(\Gamma;(R,\infty))}\mathbb {P}_{\mathfrak{C}_{n}(\Gamma)}\bigl(P',Q \bigr)u'(Q)\,d\sigma} \\ &{\quad < \epsilon.} \end{aligned}
(3.11)

Here note that $$H_{u}(P;\mathfrak{C}_{n}(\Gamma;(0,j)))$$ is the least harmonic majorant of u on $$\mathfrak{C}_{n}(\Gamma;(0,j))$$ (see [9], Theorem 3.15). If h is a harmonic majorant of u on $$\mathfrak{C}_{n}(\Gamma)$$, then

$$H_{u}\bigl(P';\mathfrak{C}_{n}\bigl( \Gamma;(0,j)\bigr)\bigr)\leq h\bigl(P'\bigr).$$

Thus we obtain from (3.11)

$$\mathbb{PI}_{\mathfrak{C}_{n}(\Gamma)}\bigl[u'\bigr]\bigl(P' \bigr)< h\bigl(P'\bigr)+\epsilon,$$

which gives the conclusion of Lemma 5. □

## 4 Proof of Theorem 1

Let P be any point of $$\mathfrak{C}_{n}(\Gamma)$$ and ϵ be any positive number. By the Vitali-Carathéodory theorem with respect to the Schrödinger operator (see [10], p.56), there exists a lower semi-continuous function $$g'(Q)$$ on $$\partial{\mathfrak {C}_{n}(\Gamma)}$$ such that

$$u'(Q)\leq g'(Q)$$
(4.1)

and

$$\mathbb{PI}_{\mathfrak{C}_{n}(\Gamma)}\bigl[g'\bigr](P)< \mathbb{PI}_{\mathfrak {C}_{n}(\Gamma)}\bigl[u'\bigr](P)+\epsilon.$$
(4.2)

Since

$$\lim_{P\in\mathfrak{C}_{n}(\Gamma), P\rightarrow Q}u(P)\leq u'(Q)\leq g'(Q)$$

for any $$Q\in\partial{\mathfrak{C}_{n}(\Gamma)}$$ from (4.1), it follows from [1], Lemma 2.1, that the limits $$\mathscr {U}_{u}$$ and $$\mathscr{U}_{u}$$ exist, and if $$-1\leq\mathscr{U}_{u}<1$$ and $$-1\leq\mathscr{V}_{u}<1$$, then

$$u(P)\leq\mathbb{PI}_{\mathfrak{C}_{n}(\Gamma)}\bigl[g'\bigr](P)+ \mathscr {V}_{u}r^{\aleph^{-}}\varphi(\Theta)+\mathscr{U}_{u}r^{\aleph ^{+}} \varphi(\Theta).$$
(4.3)

Hence we see from (4.2) and (4.3) that (2.1) holds.

Next we call the least harmonic majorant of u on $$\mathfrak{C}_{n}(\Gamma)$$: $$h_{u}(P)$$. Set $$h''(P)$$ is a Schrödinger harmonic function in $$\mathfrak{C}_{n}(\Gamma)$$ such that (see [7])

$$u(P)\leq h''(P)+\epsilon.$$
(4.4)

Put

$$h^{\ast}(P)=h_{u}(P)-h''(P)\quad\mbox{on }\mathfrak{C}_{n}(\Gamma).$$

It is easy to see that

$$h^{\ast}(P)\leq h_{u}(P).$$

It follows from Theorem B that $$\mathscr{V}_{{h^{\ast}}^{+}}<+\infty$$. Further, from Lemma 5 we see that

$$\limsup_{P\in\mathfrak{C}_{n}(\Gamma), P\rightarrow Q}h^{\ast}(P)=\liminf_{P\in\mathfrak{C}_{n}(\Gamma), P\rightarrow Q} \bigl\{ \mathbb{PI}_{\mathfrak{C}_{n}(\Gamma)}\bigl[u'\bigr](P)-h''(P) \bigr\} \leq-1.$$

From Theorem B and (4.4) we know

$$\mathscr{V}_{h^{\ast}}=\mathscr{V}_{h_{u}}-\mathscr{V}_{h''}= \mathscr {V}_{u}-\mathscr{U}_{h''}\leq\mathscr{U}_{u}- \mathscr{U}_{u}=0.$$

We see from Lemma 2 that $$-1\leq h^{\ast}(P)\leq\epsilon$$ on $$\mathfrak{C}_{n}(\Gamma)$$, which shows that $$h_{u}(P)$$ is the least harmonic majorant in $$\mathfrak{C}_{n}(\Gamma)$$. Theorem 1 is proved.

## 5 Conclusion

In this article, we dealt with a theoretical question raised in connection with the application of Dirichlet-Sch type inequality. Additionally, we discussed a particular case of it in more detail. As applications, we deduced the least harmonic majorant and log-concavity of extended subharmonic functions.