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 dσ (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.