In the following, we agree that \(\mathbf{N}=\{1,2,\ldots\}\), \(p>1\), \(\frac{1}{p}+\frac{1}{q}=1\), \(\alpha,\beta\in(0,\pi)\), \(\lambda _{1},\lambda _{2}>-\eta\), \(\lambda_{1}+\lambda_{2}=\lambda\), and for \(|x|,|y|>0\),
$$ k(x,y):=\frac{(\min\{|x|+x\cos\alpha,|y|+y\cos\beta\})^{\eta }}{(\max \{|x|+x\cos\alpha,|y|+y\cos\beta\})^{\lambda+\eta}}. $$
(5)
Lemma 1
(cf. [24])
Suppose that
\(g(t)\) (>0) is decreasing in
\(\mathbf{R}_{+}\)
and strictly decreasing in
\([n_{0},\infty)\) (\(n_{0}\in \mathbf{N}\)), satisfying
\(\int_{0}^{\infty}g(t)\,dt\in\mathbf{R}_{+}\). We have
$$ \int_{1}^{\infty}g(t)\,dt< \sum _{n=1}^{\infty}g(n)< \int_{0}^{\infty}g(t)\,dt. $$
(6)
Definition 1
Define the following weight coefficients:
$$\begin{aligned}& \omega(\lambda_{2},m) : =\sum _{|n|=1}^{\infty}k(m,n)\frac{(|m|+m\cos \alpha)^{\lambda_{1}}}{(|n|+n\cos\beta)^{1-\lambda_{2}}},\quad |m|\in \mathbf{N}, \end{aligned}$$
(7)
$$\begin{aligned}& \varpi(\lambda_{1},n) : =\sum _{|m|=1}^{\infty}k(m,n)\frac{(|n|+n\cos \beta)^{\lambda_{2}}}{(|m|+m\cos\alpha)^{1-\lambda_{1}}},\quad |n|\in \mathbf{N}, \end{aligned}$$
(8)
where \(\sum_{|j|=1}^{\infty}\cdots=\sum_{j=-1}^{-\infty}\cdots +\sum_{j=1}^{\infty}\cdots\) (\(j=m,n\)).
Lemma 2
If
\(\lambda_{2}\leq1-\eta\), then for
\(k_{\beta }(\lambda_{1}):=\frac{2(\lambda+2\eta)\csc^{2}\beta}{(\lambda _{1}+\eta )(\lambda_{2}+\eta)}\), we have
$$ k_{\beta}(\lambda_{1}) \bigl(1-\theta( \lambda_{2},m)\bigr)< \omega(\lambda _{2},m)< k_{\beta}( \lambda_{1}),\quad |m|\in\mathbf{N}, $$
(9)
where
$$\begin{aligned} \theta(\lambda_{2},m) :=&\frac{(\lambda_{1}+\eta)(\lambda _{2}+\eta)}{\lambda+2\eta} \int_{0}^{\frac{1+\cos\beta}{|m|+m\cos\alpha}}\frac{ (\min\{1,u\})^{\eta}u^{\lambda_{2}-1}}{(\max\{1,u\})^{\lambda+\eta}}\,du \\ =&O \biggl( \frac{1}{(|m|+m\cos\alpha)^{\eta+\lambda_{2}}} \biggr) \in (0,1),\quad |m|\in\mathbf{N}. \end{aligned}$$
(10)
Proof
For \(|x|>0\), we set
$$\begin{aligned}& k^{(1)}(x,y) := \frac{[\min\{|x|+x\cos\alpha,y(\cos\beta-1)\} ]^{\eta}}{[\max\{|x|+x\cos\alpha,y(\cos\beta-1)\}]^{\lambda+\eta}},\quad y< 0, \\& k^{(2)}(x,y) := \frac{[\min\{|x|+x\cos\alpha,y(1+\cos\beta)\} ]^{\eta}}{[\max\{|x|+x\cos\alpha,y(1+\cos\beta)\}]^{\lambda+\eta}},\quad y>0, \end{aligned}$$
from which we have
$$k^{(1)}(x,-y)=\frac{[\min\{|x|+x\cos\alpha,y(1-\cos\beta)\}]^{\eta }}{[\max\{|x|+x\cos\alpha,y(1-\cos\beta)\}]^{\lambda+\eta}},\quad y>0. $$
We obtain
$$\begin{aligned} \omega(\lambda_{2},m) =&\sum _{n=-1}^{-\infty}k^{(1)}(m,n)\frac{(|m|+m\cos\alpha)^{\lambda_{1}}}{[n(\cos\beta-1)]^{1-\lambda_{2}}} \\ &{}+\sum_{n=1}^{\infty}k^{(2)}(m,n) \frac{(|m|+m\cos\alpha)^{\lambda _{1}}}{[n(1+\cos\beta)]^{1-\lambda_{2}}} \\ =&\frac{(|m|+m\cos\alpha)^{\lambda_{1}}}{(1-\cos\beta)^{1-\lambda _{2}}}\sum_{n=1}^{\infty} \frac{k^{(1)}(m,-n)}{n^{1-\lambda_{2}}} \\ &{}+\frac{(|m|+m\cos\alpha)^{\lambda_{1}}}{(1+\cos\beta)^{1-\lambda _{2}}}\sum_{n=1}^{\infty} \frac{k^{(2)}(m,n)}{n^{1-\lambda_{2}}}. \end{aligned}$$
(11)
For fixed \(|m|\in\mathbf{N}\), \(\lambda_{2}\leq1-\eta\), we find that
$$\begin{aligned} \frac{k^{(1)}(m,-y)}{y^{1-\lambda_{2}}} =&\frac{[\min\{|m|+m\cos \alpha ,y(1-\cos\beta)\}]^{\eta}}{y^{1-\lambda_{2}}[\max\{|m|+m\cos\alpha ,y(1-\cos\beta)\}]^{\lambda+\eta}} \\ =& \left\{ \textstyle\begin{array}{@{}l@{\quad}l@{}} \frac{(1-\cos\beta)^{\eta}}{(|m|+m\cos\alpha)^{\lambda +\eta}}\frac{1}{y^{1-(\lambda_{2}+\eta)}},&0< y< \frac{|m|+m\cos\alpha}{1-\cos \beta} , \\ \frac{(|m|+m\cos\alpha)^{\eta}}{(1-\cos\beta)^{\lambda +\eta}}\frac{1}{y^{1+(\lambda_{1}+\eta)}},&y\geq\frac{|m|+m\cos\alpha }{1-\cos \beta}\end{array}\displaystyle \right . \end{aligned}$$
is decreasing for \(y>0\) and strictly decreasing for \(y\geq\frac {|m|+m\cos \alpha}{1-\cos\beta}\). Under the same assumptions, it is evident that
$$\begin{aligned} \frac{k^{(2)}(m,y)}{y^{1-\lambda_{2}}} =&\frac{[\min\{|m|+m\cos \alpha ,y(1+\cos\beta)\}]^{\eta}}{y^{1-\lambda_{2}}[\max\{|m|+m\cos\alpha ,y(1+\cos\beta)\}]^{\lambda+\eta}} \\ =& \left\{ \textstyle\begin{array}{@{}l@{\quad}l@{}} \frac{(1+\cos\beta)^{\eta}}{(|m|+m\cos\alpha)^{\lambda +\eta}}\frac{1}{y^{1-(\lambda_{2}+\eta)}},&0< y< \frac{|m|+m\cos\alpha}{1+\cos \beta} , \\ \frac{(|m|+m\cos\alpha)^{\eta}}{(1+\cos\beta)^{\lambda +\eta}}\frac{1}{y^{1+(\lambda_{1}+\eta)}},&y\geq\frac{|m|+m\cos\alpha }{1+\cos \beta}\end{array}\displaystyle \right . \end{aligned}$$
is decreasing for \(y>0\) and strictly decreasing for \(y\geq\frac {|m|+m\cos \alpha}{1+\cos\beta}\).
By (11) and (6), we have
$$\begin{aligned} \omega(\lambda_{2},m) < &\frac{(|m|+m\cos\alpha)^{\lambda _{1}}}{(1-\cos \beta)^{1-\lambda_{2}}} \int_{0}^{\infty}\frac {k^{(1)}(m,-y)}{y^{1-\lambda _{2}}}\,dy \\ &{}+\frac{(|m|+m\cos\alpha)^{\lambda_{1}}}{(1+\cos\beta)^{1-\lambda _{2}}} \int_{0}^{\infty}\frac{k^{(2)}(m,y)}{y^{1-\lambda_{2}}}\,dy. \end{aligned}$$
Setting \(u=\frac{y(1-\cos\beta)}{|m|+m\cos\alpha}(\frac{y(1+\cos \beta)}{|m|+m\cos\alpha})\) in the above first (second) integral, by simplifications, we find
$$\begin{aligned} \omega(\lambda_{2},m) < & \biggl(\frac{1}{1-\cos\beta}+ \frac{1}{1+\cos \beta} \biggr) \int_{0}^{\infty}\frac{(\min\{1,u\})^{\eta}u^{\lambda _{2}-1}}{(\max\{1,u\})^{\lambda+\eta}}\,du \\ =&2\csc^{2}\beta \biggl( \int_{0}^{1}u^{\eta+\lambda _{2}-1}\,du+ \int_{1}^{\infty}\frac{u^{\lambda_{2}-1}}{u^{\lambda+\eta }}\,du \biggr) \\ =&\frac{2(\lambda+2\eta)\csc^{2}\beta}{(\lambda_{1}+\eta)(\lambda _{2}+\eta)}=k_{\beta}(\lambda_{1}). \end{aligned}$$
Still by (11) and (6), we have
$$\begin{aligned} \omega(\lambda_{2},m) >&\frac{(|m|+m\cos\alpha)^{\lambda _{1}}}{(1-\cos \beta)^{1-\lambda_{2}}} \int_{1}^{\infty}\frac {k^{(1)}(m,-y)}{y^{1-\lambda _{2}}}\,dy \\ &{}+\frac{(|m|+m\cos\alpha)^{\lambda_{1}}}{(1+\cos\beta)^{1-\lambda _{2}}} \int_{1}^{\infty}\frac{k^{(2)}(m,y)}{y^{1-\lambda_{2}}}\,dy \\ \geq&\frac{1}{1-\cos\beta} \int_{\frac{1+\cos\beta}{|m|+m\cos \alpha}}^{\infty}\frac{(\min\{1,u\})^{\eta}u^{\lambda_{2}-1}}{(\max \{1,u\})^{\lambda+\eta}}\,du \\ &{}+\frac{1}{1+\cos\beta} \int_{\frac{1+\cos\beta}{|m|+m\cos\alpha}}^{\infty}\frac{(\min\{1,u\})^{\eta}u^{\lambda_{2}-1}}{(\max \{1,u\})^{\lambda+\eta}}\,du \\ =&k_{\beta}(\lambda_{1}) \bigl(1-\theta( \lambda_{2},m)\bigr)>0. \end{aligned}$$
We obtain for \(|m|+m\cos\alpha\geq1+\cos\beta\)
$$\begin{aligned} 0 < &\theta(\lambda_{2},m)=\frac{(\lambda_{1}+\eta)(\lambda _{2}+\eta)}{\lambda+2\eta} \int_{0}^{\frac{1+\cos\beta}{|m|+m\cos\alpha}}\frac{ (\min\{1,u\})^{\eta}u^{\lambda_{2}-1}}{(\max\{1,u\})^{\lambda+\eta}}\,du \\ =&\frac{(\lambda_{1}+\eta)(\lambda_{2}+\eta)}{\lambda+2\eta} \int _{0}^{\frac{1+\cos\beta}{|m|+m\cos\alpha}}u^{\eta+\lambda_{2}-1}\,du \\ =&\frac{\lambda_{1}+\eta}{\lambda+2\eta} \biggl( \frac{1+\cos\beta }{|m|+m\cos\alpha} \biggr) ^{\eta+\lambda_{2}}. \end{aligned}$$
Then we have (9) and (10). □
In the same way, we have the following.
Lemma 3
If
\(\lambda_{1}\leq1-\eta\), then for
\(k_{\alpha }(\lambda_{1})=\frac{2(\lambda+2\eta)\csc^{2}\alpha}{(\lambda _{1}+\eta )(\lambda_{2}+\eta)}\), we have
$$ k_{\alpha}(\lambda_{1}) \bigl(1-\vartheta( \lambda_{1},n)\bigr)< \varpi(\lambda _{1},n)< k_{\alpha}( \lambda_{1}),\quad |n|\in\mathbf{N}, $$
(12)
where
$$\begin{aligned} \vartheta(\lambda_{1},n) :=&\frac{(\lambda_{1}+\eta)(\lambda _{2}+\eta)}{\lambda+2\eta} \int_{0}^{\frac{1+\cos\alpha}{|n|+n\cos\beta }}\frac{(\min\{1,u\})^{\eta}u^{\lambda_{1}-1}}{(\max\{1,u\})^{\lambda+\eta}}\,du \\ =&O \biggl( \frac{1}{(|n|+n\cos\beta)^{\eta+\lambda_{1}}} \biggr) \in (0,1),\quad |n|\in\mathbf{N}. \end{aligned}$$
(13)
Lemma 4
If
\(\theta\in(0,\pi)\), then for
\(\rho>0\), \(H_{\rho }(\theta):=\sum_{|n|=1}^{\infty}\frac{1}{(|n|+n\cos\theta)^{1+\rho}}\), we have
$$ H_{\rho}(\theta)= \biggl[ \frac{1}{(1+\cos\theta)^{1+\rho}}+ \frac{1}{ (1-\cos\theta)^{1+\rho}} \biggr] \frac{1+\rho O(1)}{\rho} \quad\bigl(\rho \rightarrow0^{+}\bigr). $$
(14)
Proof
We have
$$\begin{aligned} H_{\rho}(\theta) =&\sum_{n=-1}^{-\infty} \frac{1}{[n(\cos\theta -1)]^{1+\rho}}+\sum_{n=1}^{\infty} \frac{1}{[n(\cos\theta +1)]^{1+\rho}} \\ =& \biggl[ \frac{1}{(1-\cos\theta)^{1+\rho}}+\frac{1}{(1+\cos\theta )^{1+\rho}} \biggr] \sum _{n=1}^{\infty}\frac{1}{n^{1+\rho}}. \end{aligned}$$
By (6), we find
$$\begin{aligned}& \begin{aligned} H_{\rho}(\theta) &= \biggl[ \frac{1}{(1-\cos\theta)^{1+\rho}}+ \frac {1}{(1+\cos\theta)^{1+\rho}} \biggr] \Biggl( 1+\sum_{n=2}^{\infty} \frac {1}{n^{1+\rho}} \Biggr) \\ &< \biggl[ \frac{1}{(1-\cos\theta)^{1+\rho}}+\frac{1}{(1+\cos\theta )^{1+\rho}} \biggr] \biggl( 1+ \int_{1}^{\infty}\frac{dy}{y^{1+\rho }} \biggr) \\ &=\frac{1}{\rho} \biggl[ \frac{1}{(1-\cos\theta)^{1+\rho}}+\frac{1}{(1+\cos\theta)^{1+\rho}} \biggr] (1+ \rho), \end{aligned} \\& \begin{aligned} H_{\rho}(\theta) &> \biggl[ \frac{1}{(1-\cos\theta)^{1+\rho}}+\frac {1}{(1+\cos\theta)^{1+\rho}} \biggr] \int_{1}^{\infty}\frac {dy}{y^{1+\rho}} \\ &= \frac{1}{\rho} \biggl[ \frac{1}{(1-\cos\theta)^{1+\rho}}+\frac{1}{(1+\cos\theta)^{1+\rho}} \biggr] . \end{aligned} \end{aligned}$$
Hence we have (14). □