1 Introduction and preliminary concepts

In 1951, Fast [2] and Steinhaus [3] independently introduced the concept of statistical convergence for sequences of real numbers, and since then this concept has been generalized and investigated in different ways by different authors. Likewise summability theory and convergence of sequences have also been studied actively in the area of pure mathematics for the last several decades. Extensive works on the topic are applicable in topology, functional analysis, Fourier analysis, measure theory, applied mathematics, mathematical modeling, computer science, analytic number theory, etc. One may refer to [4,5,6,7,8,9], etc.

Let \(A \subseteq \mathbb{N}\) and \(A_{n}=\{\psi \leq n: \psi \in A \}\). We say that A has natural density, i.e., \(\delta (A)= \lim_{n} \frac{1}{n} \vert A_{n} \vert \), if the limit exists, where \(\vert A_{n} \vert \) denotes the cardinality of \(A_{n}\).

By the concept of statistical convergence, we mean a sequence \(\tilde{x}=(x_{\psi })\) of real numbers which statistically converges to ℓ if for every \(\varepsilon >0\) the set \(A_{\varepsilon }= \{ \psi \in \mathbb{N}:\vert x_{\psi }- \ell \vert \geq \varepsilon \}\) has natural density zero, i.e., for each \(\varepsilon >0\),

$$ \lim_{n} \frac{1}{n} \bigl\vert \bigl\{ \psi \leq n : \vert x_{\psi }-\ell \vert \geq \varepsilon \bigr\} \bigr\vert =0. $$

Let \(\lambda =(\lambda _{n})\) be a non-decreasing sequence of positive numbers tending to ∞ such that

$$ \lambda _{n+1} \leq \lambda _{n}+1 $$
(1)

and \(\lambda _{1}=0\).

The generalized de la Vallee Poussin mean of a sequence \(\tilde{x}=(x _{\psi })\) is defined by \(T_{n}(x)=\frac{1}{\lambda _{n}} \sum_{\psi \in I_{n}} x_{\psi }\), where \(I_{n}=[n-\lambda _{n}+1,n]\).

Now, a sequence \(\tilde{x}=(x_{\psi })\) is said to be \((V,\lambda )\)-summable to ℓ if \(T_{n}(x)\) converges to ℓ, i.e.,

$$ \lim_{n} \frac{1}{\lambda _{n}} \sum_{\psi \in I_{n}} \vert x_{\psi }-\ell \vert =0. $$

Also a sequence \(\tilde{x}=(x_{\psi })\) is said to be statistically λ-convergent to ℓ if, for every \(\varepsilon >0\),

$$ \lim_{n} \frac{1}{n} \bigl\vert \bigl\{ \psi \leq n : \bigl\vert T_{\psi }(x)-\ell \bigr\vert \geq \varepsilon \bigr\} \bigr\vert =0. $$

By logarithmic density, we mean \(\delta _{\log _{n}}(E)=\frac{1}{\log _{n}} \sum_{\psi =1}^{n} \frac{\chi _{E}(\psi )}{\psi }\) for \(E \in \mathbb{N}\), where \(\log _{n}= \sum_{\psi =1}^{n} \frac{1}{\psi } \approx \log n\), \(n \in \mathbb{N}\) [8].

A sequence \(\tilde{x}=(x_{\psi })\) is logarithmic statistically convergent to ℓ if

$$ \lim_{n} \frac{1}{\log _{n}} \biggl\vert \biggl\{ \psi \leq n : \frac{1}{ \psi } \vert x_{\psi }-\ell \vert \geq \varepsilon \biggr\} \biggr\vert =0. $$

A sequence \(\tilde{x}=(x_{\psi })\) is logarithmic \((V,\lambda )\)-statistically convergent to ℓ if

$$ \lim_{n} \frac{1}{\log _{\lambda _{n}}} \biggl\vert \biggl\{ \psi \in I_{n}: \frac{1}{\psi } \bigl\vert T_{\psi }(x)-\ell \bigr\vert \geq \varepsilon \biggr\} \biggr\vert =0, $$

where \(\log _{\lambda _{n}}= \sum_{\psi =1}^{\lambda _{n}} \frac{1}{\psi } \approx \log \lambda _{n}\) (\(n=1,2,3,\ldots\)).

Let \(\mu _{n} =\frac{1}{\log _{\lambda _{n}}} \sum_{\psi \in I_{n}} \frac{T_{\psi }(x)}{\psi }\), where \(\log _{\lambda _{n}}= \sum_{\psi =1}^{\lambda _{n}} \frac{1}{\psi } \approx \log \lambda _{n} \) (\(n=1,2,3,\ldots\)). A sequence \(\tilde{x}=(x_{\psi })\) is logarithmic \((V,\lambda )\)-summable to ℓ if \((\mu _{n})\) is convergent to ℓ, i.e., \(\lim_{n} \frac{1}{\log _{\lambda _{n}}} \sum_{\psi \in I_{n}} \frac{ \vert T_{\psi }(x) -\ell \vert }{\psi } =0\).

A sequence \(\tilde{x}=(x_{\psi })\) is logarithmic \((V,\lambda )\)-statistically summable to ℓ if \((\mu _{n})\) is λ-statistically convergent, i.e.,

$$ \lim_{n} \frac{1}{\lambda _{n}} \bigl\vert \bigl\{ \psi \in I_{n}: \vert \mu _{ \psi }-\ell \vert \geq \varepsilon \bigr\} \bigr\vert =0. $$

We define it as \(st_{\log _{\lambda _{n}}}- \lim_{n} T_{n}=\ell \).

Moricz [10] studied the concept of Tauberian conditions for statistical convergence followed from statistical summability \((C,1)\). Braha [11] extended these results using Tauberian conditions for λ-statistical convergence, which was followed from statistical summability \((V,\lambda )\). Braha [12] also explained the Tauberian theorems for the generalized Norlund–Euler summability method. One may refer to [13,14,15].

In this paper, we study the Tauberian theorems for logarithmic \((V,\lambda )\)-statistical convergence which is followed from de la Vallee Poussin mean. We also try to establish some results involving the logarithmic density.

2 Main results

Theorem 2.1

Let λ be a real-valued sequence defined in (1). Then,

  1. 1.

    If \(\tilde{x}=(x_{\psi })\) is logarithmic \((V,\lambda )\)-statistically summable to ℓ, then it is logarithmic \((V,\lambda )\)-statistically convergent to ℓ, provided \(\lim \inf_{n} \frac{1}{\lambda _{n}}>0\).

  2. 2.

    If \(\tilde{x}=(x_{\psi })\) is bounded, then logarithmic \((V,\lambda )\)-statistical convergence implies logarithmic \((V,\lambda )\)-statistical summability.

  3. 3.

    \(\varOmega (\log _{n},\lambda ) \cap \ell _{\infty }=\varPi (\log _{n},\lambda )\), where \(\varOmega (\log _{n},\lambda ) \) is the collection of all logarithmic \((V,\lambda )\)-statistical convergence sequences, \(\ell _{\infty }\) is the collection of all bounded sequences, and \(\varPi (\log _{n},\lambda )\) is the collection of all logarithmic \((V,\lambda )\)-summable sequences.

Proof

(1) Let \(\tau _{n}= \{ \psi \in I_{n}: \frac{1}{\log _{\lambda _{n}}} \sum_{\psi \in I_{n}} \frac{1}{\psi } \vert T_{\psi }(x)-\ell \vert \geq \varepsilon \} \).

Since \(\tilde{x}=(x_{\psi })\) is logarithmic \((V,\lambda )\)-statistically summable to ℓ, then \(\tau _{n}\) is λ-statistically convergent to ℓ, i.e.,

$$ \lim_{n} \frac{1}{\lambda _{n}} \biggl\vert \biggl\{ \psi \in I_{n}: \frac{1}{ \log _{\lambda _{n}}} \sum_{\psi \in I_{n}} \frac{1}{\psi } \bigl\vert T_{\psi }(x)-\ell \bigr\vert \geq \varepsilon \biggr\} \biggr\vert =0. $$

Also we can write

$$\begin{aligned} \frac{1}{\log _{\lambda _{n}}} \sum_{\psi \in I_{n}} \frac{1}{\psi } \bigl\vert T_{\psi }(x) - \ell \bigr\vert \geq & \frac{1}{\log _{\lambda _{n}}} \sum _{\psi \in I_{n}, \vert \frac{T_{\psi }(x)}{\psi }-\ell \vert \geq \varepsilon } \frac{1}{\psi } \bigl\vert T_{\psi }(x) -\ell \bigr\vert \\ \geq & \frac{1}{\log _{\lambda _{n}}} \biggl( \biggl\vert \biggl\{ \psi \in I_{n}: \frac{1}{\psi } \bigl\vert T_{\psi }(x)-\ell \bigr\vert \geq \varepsilon \biggr\} \biggr\vert \varepsilon \biggr), \end{aligned}$$

which implies that

$$\begin{aligned} \frac{1}{\lambda _{n}} \biggl\{ \psi \in I_{n}: \frac{1}{ \log _{\lambda _{n}}} \sum _{\psi \in I_{n}} \frac{1}{\psi } \bigl\vert T_{\psi }(x) - \ell \bigr\vert \geq \varepsilon \biggr\} \geq & \frac{1}{\lambda _{n}} \frac{ ( \vert \{ \psi \in I_{n}: \frac{1}{\psi } \vert T _{\psi }(x) -\ell \vert \geq \varepsilon \} \vert \varepsilon )}{\log _{\lambda _{n}} }. \end{aligned}$$

Since \(\lim \inf_{n} \frac{1}{\lambda _{n}} >0\) and \(\tilde{x}=(x_{\psi })\) is logarithmic \((V,\lambda )\)-statistically summable to ℓ, so by taking \(n \rightarrow \infty \), we get \(\tilde{x}=(x_{\psi })\) is logarithmic \((V,\lambda )\)-statistically convergent to ℓ. This completes the proof. □

Proof

(2) Let \(\tilde{x}=(x_{\psi })\) be bounded and logarithmic \((V,\lambda )\)-statistically convergent to ℓ. Then there exists \(M>0\) such that \(\vert x_{\psi }-\ell \vert \leq M\). Now, for any \(\varepsilon >0\),

$$\begin{aligned} \frac{1}{\log _{\lambda _{n}}} \sum_{\psi \in I_{n}} \frac{1}{\psi } \bigl\vert T_{\psi }(x) - \ell \bigr\vert =& \frac{1}{\log _{\lambda _{n}}} \sum _{\psi \in I_{n}, \psi \notin B(n) } \frac{1}{\psi } \bigl\vert T_{ \psi }(x) - \ell \bigr\vert \\ &{}+ \frac{1}{\log _{\lambda _{n}}} \sum_{\psi \in I_{n}, \psi \in B(n)} \frac{1}{\psi } \bigl\vert T_{\psi }(x) - \ell \bigr\vert \\ =& K_{1}(n) + K_{2}(n), \end{aligned}$$

where \(B(n)= \{\psi \in I_{n} : \frac{1}{\psi } \vert T_{\psi }(x)- \ell \vert \geq \varepsilon \}\)

Now, if \(\psi \notin B(n)\), then \(K_{1}(n) < \varepsilon \). For \(\psi \in B(n)\), we have

$$ K_{2}(n) \leq \bigl(\sup \bigl\vert T_{\psi }(x)-\ell \bigr\vert \bigr). \bigl( \bigl\vert B(n) \bigr\vert /\log _{\lambda _{n}} \bigr) \leq M \bigl\vert B(n) \bigr\vert / \log _{\lambda _{n}} \rightarrow 0 $$

as \(n \rightarrow \infty \).

Since logarithmic density of \(B(n)\) is zero, hence we can say that \(\tilde{x}=(x_{\psi })\) is logarithmic \((V,\lambda )\)-statistically summable. This completes the proof. □

Proof

Proof of (3) follows from the proof of (1) and (2), so it is omitted here. □

3 Tauberian theorems

Theorem 3.1

Let \((\lambda _{n})\) be a sequence of real numbers and \(st_{ \log _{\lambda _{n}}}- \lim_{n} \inf \frac{\lambda _{t_{n}}}{\lambda _{n}} >1 \) for all \(t>1\), where \(t_{n}\) denotes the integral parts of \([t.n]\) for every \(n \in \mathbb{N}\), and let \((T_{\psi })\) be a sequence of real numbers such that \(st_{\log _{\lambda _{n}}}- \lim_{n} T_{n} =\ell \). Then \(\tilde{x}=(x_{\psi })\) is \(st_{ \log _{\lambda _{n}}}\)-convergent to ℓ iff the following conditions hold:

$$ \inf_{t>1} \lim_{n} \sup \frac{1}{\log _{\lambda _{n}}} \Biggl\vert \Biggl\{ \psi \in I_{n}: \frac{1}{\lambda _{t_{\psi }}-\lambda _{\psi }} \sum _{j=\psi +1}^{t_{\psi }} \frac{1}{\psi }(x_{j}-x_{\psi }) \leq - \varepsilon \Biggr\} \Biggr\vert =0 $$

and

$$ \inf_{0< t< 1} \lim_{n} \sup \frac{1}{\log _{\lambda _{n}}} \Biggl\vert \Biggl\{ \psi \in I_{n}: \frac{1}{\lambda _{\psi }-\lambda _{t_{\psi }}} \sum _{j=t_{\psi }+1}^{k} \frac{1}{\psi } (x_{\psi }-x_{j}) \leq - \varepsilon \Biggr\} \Biggr\vert =0. $$

Remark

Let us suppose that

$$ st_{\log _{\lambda _{n}}}- \lim_{n} x_{n} =\ell \quad \text{and}\quad st_{\log _{\lambda _{n}}}- \lim_{n} T_{n} =\ell $$
(2)

are satisfied, then for every \(t>1\), the following relation is valid:

$$st_{\log _{\lambda _{n}}}- \lim_{n} x_{n} =\ell \quad \mbox{implies that} \quad \lim_{n} \frac{1}{\log _{\lambda _{n}}} \sum _{\psi =1}^{n} \frac{1}{\psi } \vert x_{\psi }-\ell \vert =0 $$

and

$$st_{\log _{\lambda _{n}}}- \lim_{n} T_{n} =\ell \quad \mbox{implies that} \quad \lim_{n} \frac{1}{\log _{\lambda _{n}}} \sum _{\psi =1}^{n} \frac{1}{\psi } \bigl\vert T_{\psi }(x)-\ell \bigr\vert =0, $$

from which it follows that \(st_{\lambda }- \lim_{n} \frac{1}{\log _{(\lambda _{t_{\psi }}-\lambda _{\psi })}} \sum_{\psi =n+1}^{t_{n}} \frac{1}{\psi } x_{\psi }=0\) holds for \(t>1\), i.e.,

$$ \lim_{n} \frac{1}{\log _{\lambda _{n}}} \Biggl\vert \Biggl\{ \psi \leq n: \frac{1}{\lambda _{t_{\psi }}-\lambda _{\psi }} \sum_{\psi =1}^{t_{n}} \frac{ \vert x_{\psi } \vert }{\psi } \geq \varepsilon \Biggr\} \Biggr\vert =0, $$

and for \(0< t<1\), we have \(st_{\lambda }- \lim_{n} \frac{1}{\lambda _{\psi }-\lambda _{t_{\psi }}} \sum_{\psi =t_{n}+1}^{n} \frac{x_{\psi }}{\psi }=0\), i.e.,

$$ \lim_{n} \frac{1}{\log _{\lambda _{n}}} \Biggl\vert \Biggl\{ \psi \leq n: \frac{1}{\lambda _{\psi }-\lambda _{t_{\psi }}} \sum_{\psi =t_{n}+1}^{n} \frac{ \vert x_{\psi } \vert }{\psi } \geq \varepsilon \Biggr\} \Biggr\vert =0 $$

holds.

Lemma 3.1

For the sequence of real numbers \(\lambda =(\lambda _{n})\), (2) is equivalent to \(st_{\lambda _{n}}- [4] \lim_{n} \inf \frac{\lambda _{n}}{\lambda _{t_{n}}}>1\) for all \(0< t<1\) [12].

Lemma 3.2

If \(st_{\log _{\lambda _{n}}}- \lim_{n} x_{n} =\ell \) and \(st_{\log _{\lambda _{n}}}- \lim_{n} T_{n} =\ell \) are satisfied, and let \(\tilde{x}=(x_{\psi })\) be a sequence of complex numbers which is logarithmic \((V,\lambda )\)-statistically convergent to ℓ, then for any \(t>1\),

$$st_{\log _{\lambda _{n}}}- \lim_{n} T_{t_{n}}=\ell ,\quad \textit{i.e.}, \quad \frac{1}{\log _{\lambda _{n}}} \Biggl\vert \Biggl\{ \psi \leq n : \sum _{\psi =1}^{n} \frac{1}{\psi } \vert T_{t_{\psi }}- \ell \vert \geq \varepsilon \Biggr\} \Biggr\vert =0. $$

Proof

Case I: Let us consider that \(t >1\), then from construction of the sequence \(\lambda =(\lambda _{n})\) we get

$$ \lim_{n} (n -\lambda _{n})= \lim _{n} (t_{n}-\lambda _{t_{n}}), $$
(3)

and for every \(\varepsilon >0\), we have

$$\begin{aligned}& \biggl\{ \psi \in I_{t_{n}}:\frac{1}{\psi } \vert T_{t_{\psi }}- \ell \vert \geq \varepsilon \biggr\} \\& \quad \subset\biggl\{ \psi \in I_{n} : \frac{1}{\psi } \vert T_{\psi }-L \vert \geq \varepsilon \biggr\} \cup \Biggl\{ \psi \in I_{n}: \frac{1}{ \log _{\lambda _{\psi }}} \sum_{j=\psi -\lambda _{\psi }+1}^{\psi } \frac{x_{j}}{j} \neq \frac{1}{ \log _{\lambda _{t_{\psi }}}} \sum_{j=t_{\psi }-\lambda _{t_{\psi }}+1}^{t_{\psi }} \frac{x_{j}}{j} \Biggr\} . \end{aligned}$$

Following Eq. (3), we can say that \(st_{\log _{\lambda }}- \lim T_{t_{n}}=\ell \).

Case II: Now suppose that \(0< t<1\). For the definition of \(t_{n}=[t.n]\), for any natural number n, we can conclude that \((T_{t_{n}})\) does not appear more than \([1+t^{-1}]\) times in the sequence \((T_{n})\). In fact, if there exist integers ψ, m such that

$$ n \leq t.\psi < t(\psi +1)< \cdots< t(\psi +m-1)< n+1\leq t(\psi +m), $$

then

$$ n+t(m-1) \leq t(\psi +m-1)< n+1 \quad \Rightarrow\quad m< 1+\frac{2}{t}. $$

So, we have the following inequality:

$$\begin{aligned} \frac{1}{\log _{\lambda _{t_{n}}}} \biggl\vert \biggl\{ \psi \in I_{t _{n}}: \frac{1}{\psi } \vert T_{t_{\psi }}-\ell \vert \geq \varepsilon \biggr\} \biggr\vert \leq & \biggl(1+\frac{1}{t} \biggr) \frac{1}{\log _{\lambda _{t_{n}}}} \biggl\vert \biggl\{ \psi \in I_{n} : \frac{1}{\psi } \vert T_{ \psi }-\ell \vert \geq \varepsilon \biggr\} \biggr\vert \\ \leq & 2(1+t)\frac{1}{\log _{\lambda _{n}}} \biggl\vert \biggl\{ \psi \in I_{n} :\frac{1}{\psi } \vert T_{\psi }-\ell \vert \geq \varepsilon \biggr\} \biggr\vert , \end{aligned}$$

which gives that \(st_{\log _{\lambda _{n}}}- \lim_{n} T_{t_{n}}=\ell \). □

Lemma 3.3

If \(st_{\log _{\lambda _{n}}}- \lim_{n} x_{n} =\ell\) and \(st_{\log _{\lambda _{n}}}- \lim_{n} T_{n} =\ell \) are satisfied and \(\tilde{x}=(x_{\psi })\) is logarithmic \((V,\lambda )\)-statistically convergent to ℓ, then we have

$$ \mathrm{(i)}\quad \lim_{n} \frac{1}{\log _{\lambda _{n}}} \Biggl\vert \Biggl\{ \psi \leq n: \frac{1}{\lambda _{t_{n}}-\lambda _{n}} \sum_{\psi =n+1}^{t_{n}} \frac{ \vert x_{\psi }-\ell \vert }{\psi } \geq \varepsilon \Biggr\} \Biggr\vert =0 \quad \textit{for every } t>1 $$
(4)

and

$$ \mathrm{(ii)}\quad \lim_{n} \frac{1}{\log _{\lambda _{n}}} \Biggl\vert \Biggl\{ \psi \leq n: \frac{1}{\lambda _{n}-\lambda _{t_{n}}} \sum_{\psi =t_{n}+1}^{n} \frac{1}{\psi } \vert x_{\psi }-\ell \vert \geq \varepsilon \Biggr\} \Biggr\vert =0\quad \textit{for any } 0< t< 1. $$
(5)

Proof

(i) Let us suppose that \(t>1\). We get

$$\begin{aligned}& \begin{aligned}[b] \frac{1}{\lambda _{t_{n}}-\lambda _{n}} \sum_{j=n+1}^{t_{n}} \frac{x_{j}}{j}&=T_{n}+\frac{\lambda _{t_{n}}}{ \lambda _{t_{n}}-\lambda _{n}}(T_{t_{n}}-T_{n})\\ &\quad {}+ \frac{1}{\lambda _{t_{n}}- \lambda _{n}} \sum_{j=n-\lambda _{n}+1}^{t_{n}} \frac{x_{j}}{j}-\frac{1}{\lambda _{t _{n}}-\lambda _{n}} \sum_{j=t_{n}-\lambda _{t_{n}}+1}^{t_{n}} \frac{x_{j}}{j} \end{aligned} \\& \begin{aligned}[b] \quad \Rightarrow\quad \frac{1}{\lambda _{t_{n}}-\lambda _{n}} \sum_{j=n+1}^{t_{n}} \frac{x_{j}}{j} & =T_{n}+\frac{\lambda _{t_{n}}}{ \lambda _{t_{n}}-\lambda _{n}}(T_{t_{n}}-T_{n})\\ &\quad {}+ \frac{1}{\lambda _{t_{n}}- \lambda _{n}} \Biggl( \sum_{j=n-\lambda _{n}+1}^{t_{n}} \frac{x_{j}}{j}- \sum_{j=t_{n}-\lambda _{t_{n}}+1}^{t_{n}} \frac{x_{j}}{j} \Biggr). \end{aligned} \end{aligned}$$
(6)

From the definition of the sequence \((\lambda _{n})\) and logarithmic density, we obtain

$$ st_{\log _{\lambda _{n}}}- \lim_{n} \sup \sum _{j=n-\lambda _{n}+1}^{t_{n}} x_{j}=st_{\log _{\lambda _{n}}}- \lim _{n} \sup \sum_{j=t_{n}-\lambda _{t_{n}}+1}^{t_{n}} x_{j}. $$
(7)

Let us suppose that \(st_{\log _{\lambda _{n}}}- \lim_{n} \sup \sum_{j=n-\lambda _{n}+1}^{t_{n}} x_{j} =L\), and for every \(\varepsilon >0\), we get

$$\begin{aligned}& \frac{ \lim_{n} \frac{1}{\log _{\lambda _{n}}} \vert \{ \psi \in I_{t_{n}}: \vert \sum_{j=t_{\psi }-\lambda _{t_{\psi }}+1}^{t_{\psi }} \frac{ \vert x _{j}-\ell \vert }{j} \vert \geq \varepsilon \} \vert }{ \lambda _{t_{n}}} \\& \quad \leq \frac{ \lim_{n} \frac{1}{\log _{\lambda _{n}}} \vert \{ \psi \in I_{n}: \vert \sum_{j=\psi -\lambda _{\psi }+1}^{t_{\psi }} \frac{ \vert x_{j}-\ell \vert }{j} \vert \geq \varepsilon \} \vert }{ \lambda _{n}} \\& \qquad {}+\frac{ \lim_{n} \frac{1}{\log _{\lambda _{n}}} \vert \{ \psi \in I_{n}: \sum_{j=t_{\psi }-\lambda _{t_{\psi }}+1}^{t_{\psi }} \frac{x_{j}}{j} \neq \sum_{j=\psi -\lambda _{\psi }+1}^{t_{\psi }}\frac{x_{j}}{j} \} \vert }{\lambda _{n}}, \end{aligned}$$

from which it follows that \(st_{\log _{\lambda _{n}}}- \lim_{n} \sup \sum_{j=t_{n}-\lambda _{t_{n}}+1}^{t_{n}} x_{j} =L\).

Also, since \(st_{\lambda }- \lim_{n} \sup \frac{\lambda _{t_{n}}}{\lambda _{t_{n}}-\lambda _{n}} < \infty \) and \(st_{\lambda }- \lim_{n} \sup \frac{1}{\lambda _{t_{n}}-\lambda _{n}} < \infty \), then we get

$$st_{\log _{\lambda _{n}}}- \lim_{n} \frac{1}{\lambda _{t_{n}}-\lambda _{n}} \sum _{j=n+1}^{t_{n}} x_{j} =\ell . $$

(ii) If \(0< t<1\), we have

$$\frac{1}{\lambda _{n}-\lambda _{t_{n}}} \sum_{j=t_{n}+1}^{n} \frac{x_{j}}{j}=T_{n}+\frac{\lambda _{t_{n}}}{ \lambda _{n}-\lambda _{t_{n}}}(T_{n}-T_{t_{n}})+ \frac{1}{\lambda _{n}- \lambda _{t_{n}}} \sum_{j=n-\lambda _{n}+1}^{n} \frac{x_{j}}{j} -\frac{1}{\lambda _{t_{n}}- \lambda _{n}} \sum_{j=t_{n}-\lambda _{t_{n}}+1}^{n} \frac{x_{j}}{j}. $$

This completes the proof. □

Following the above procedure, we can get the proof of Theorem 3.1.

Proof of Theorem 3.1

Let us suppose that \(st_{\log _{\lambda }}- \lim_{\psi }x_{\psi }=L\) and \(st_{\log _{\lambda }}- \lim_{\psi }T_{\psi }=\ell \). For every \(t>1\), we get (by Lemma 3.2)

$$ \inf_{t>1} \lim_{n} \sup \frac{1}{\log _{\lambda _{n}}} \Biggl\vert \Biggl\{ \psi \in I_{n}: \frac{1}{\lambda _{t_{\psi }}-\lambda _{\psi }} \sum _{j=\psi +1}^{t_{\psi }} \frac{1}{\psi }(x_{j}-x_{\psi }) \leq - \varepsilon \Biggr\} \Biggr\vert =0. $$

Similarly, if \(0< t<1\), we obtain (by Lemma 3.2)

$$ \inf_{0< t< 1} \lim_{n} \sup \frac{1}{\log _{\lambda _{n}}} \Biggl\vert \Biggl\{ \psi \in I_{n}: \frac{1}{\lambda _{\psi }-\lambda _{t_{\psi }}} \sum _{j=t_{\psi }+1}^{\psi }\frac{1}{\psi } (x_{\psi }-x_{j}) \leq - \varepsilon \Biggr\} \Biggr\vert =0. $$

Now assume that \(st_{\log _{\lambda }}- \lim_{n} T_{n} =\ell \) and

$$\begin{aligned}& st_{\log _{\lambda _{n}}}- \lim_{n} \inf \frac{\lambda _{t_{n}}}{\lambda _{n}} >1 \quad \text{for all } t>1, \end{aligned}$$
(8)
$$\begin{aligned}& \inf_{t>1} \lim_{n} \sup \frac{1}{\log _{\lambda _{n}}} \Biggl\vert \Biggl\{ \psi \in I_{n}: \frac{1}{\lambda _{t_{\psi }}-\lambda _{\psi }} \sum _{j=\psi +1}^{t_{\psi }} \frac{1}{\psi }(x_{j}-x_{\psi }) \leq - \varepsilon \Biggr\} \Biggr\vert =0, \end{aligned}$$
(9)

and

$$ \inf_{0< t< 1} \lim_{n} \sup \frac{1}{\log _{\lambda _{n}}} \Biggl\vert \Biggl\{ \psi \in I_{n}: \frac{1}{\lambda _{\psi }-\lambda _{t_{\psi }}} \sum _{j=t_{\psi }+1}^{\psi }\frac{1}{\psi } (x_{\psi }-x_{j}) \leq - \varepsilon \Biggr\} \Biggr\vert =0 $$
(10)

are satisfied. We have to prove that \(st_{\log _{\lambda }}- \lim_{n} x_{n} =\ell \) or equivalently \(st_{\log _{\lambda }}- [4] \lim_{n} (T_{n}-x_{n})=0\).

Case I: If \(t>1\), let us suppose

$$ x_{n}-T_{n}=\lambda _{t_{n}}\frac{T_{t_{n}}-T_{n}}{\lambda _{t_{n}}- \lambda _{n}}- \frac{1}{\lambda _{t_{n}}-\lambda _{n}} \sum_{j=n+1}^{t_{n}} \frac{x_{j}-x_{n}}{j}. $$

For any \(\varepsilon >0\), we obtain

$$\begin{aligned}& \{\psi \in I_{n}: x_{n}-T_{n} \geq \varepsilon \} \\& \quad \subset \biggl\{ \psi \in I_{n} : \frac{\lambda _{t_{n}}}{\lambda _{t_{n}}- \lambda _{n}}(T_{t_{n}}-T_{n}) \geq \frac{\varepsilon }{2} \biggr\} \cup \Biggl\{ \psi \in I_{n} : \frac{1}{\lambda _{t_{n}}-\lambda _{n}} \sum_{j=n+1}^{t_{n}} \frac{x_{j}-x_{\psi }}{j} \leq \frac{- \varepsilon }{2} \Biggr\} . \end{aligned}$$

From the above relation (9), it follows that, for any arbitrary \(\gamma >0\), there exists \(t>1\) such that

$$ \lim_{n} \sup \frac{1}{\log _{\lambda _{n}}} \Biggl\vert \Biggl\{ \psi \in I_{n}: \frac{1}{\lambda _{t_{\psi }}-\lambda _{\psi }} \sum_{j=\psi +1}^{t_{\psi }} \frac{1}{x_{j}-x_{\psi }}{j} \leq - \varepsilon \Biggr\} \Biggr\vert \leq \gamma . $$

Also following Lemma 3.2 and the relations \(st_{\lambda }- \lim_{n} \sup \frac{\lambda _{t_{n}}}{\lambda _{t_{n}}-\lambda _{n}} < \infty \) and \(st_{\lambda }- [4] \lim_{n} \sup \frac{1}{\lambda _{t_{n}}-\lambda _{n}} < \infty \), we get

$$ \lim_{n} \sup \frac{1}{\log _{\lambda _{n}}} \biggl\vert \biggl\{ \psi \in I_{n} : \biggl\vert \frac{\lambda _{t_{n}}}{\lambda _{t_{n}}- \lambda _{n}} {T_{t_{n}}-T_{n}} \biggr\vert \geq \frac{\varepsilon }{2} \biggr\} \biggr\vert =0. $$

Combining these relations, we have

$$ \lim_{n} \sup \frac{1}{\log _{\lambda _{n}}} \biggl\vert \biggl\{ \psi \in I_{n} : \sum_{j} \frac{x_{n} -T_{n}}{j} \geq \varepsilon \biggr\} \biggr\vert \leq \gamma . $$

Since γ is arbitrary, we conclude that, for every \(\varepsilon >0\),

$$ \lim_{n} \sup \frac{1}{\log _{\lambda _{n}}} \biggl\vert \biggl\{ \psi \in I_{n}: \sum_{j} \frac{x_{n} -T_{n}}{j} \geq \varepsilon \biggr\} \biggr\vert =0. $$

Case II: If \(0< t<1\), let us suppose

$$ x_{n}-T_{n}=\lambda _{t_{n}}\frac{T_{t_{n}}-T_{n}}{\lambda _{n}- \lambda _{t_{n}}}- \frac{1}{\lambda _{n}-\lambda _{t_{n}}} \sum_{j=t_{n}+1}^{n} \frac{x_{n}-x_{j}}{j}. $$

For any \(\varepsilon >0\),

$$\begin{aligned}& \{ \psi \in I_{n}: x_{n} -T_{n} \leq -\varepsilon \} \\& \quad \subset\biggl\{ \psi \in I_{n}: \frac{\lambda _{t_{n}}}{\lambda _{n} - \lambda _{t_{n}}}(T_{n} -T_{t_{n}}) \leq -\frac{\varepsilon }{2} \biggr\} \cup \Biggl\{ \psi \in I_{n} : \frac{1}{\lambda _{n}-\lambda _{t_{n}}} \sum_{j =t_{n}+1}^{n} \frac{x_{\psi }-x_{j}}{j} \leq -\frac{\varepsilon }{2} \Biggr\} . \end{aligned}$$

Proceeding in the same way as above, we get the result as follows:

$$ \lim_{n} \frac{1}{\log _{\lambda _{n}}} \biggl\vert \biggl\{ \psi \in I_{n} : \sum_{n} \frac{x_{n}-T_{n}}{n} \geq \varepsilon \biggr\} \biggr\vert =0. $$

This completes the proof of the theorem. □

Theorem 3.2

Let \((\lambda _{n})\) be a sequence of complex numbers which satisfies the following condition:

$$ st_{\log _{\lambda _{n}}}- \lim_{n} \inf \frac{\lambda _{t_{n}}}{\lambda _{n}} >1 \quad \textit{for all } t>1, $$

and also consider that \(st_{\log _{\lambda }}-\lim T_{\psi }=\ell \). Then \((x_{\psi })\) is \(st_{\log _{\lambda }}\)-statistically convergent to the same number ℓ if and only if the following two conditions hold: for every \(\varepsilon >0\),

$$ \inf_{t>1} \lim_{n} \sup \frac{1}{\log _{\lambda _{n}}} \Biggl\vert \Biggl\{ \psi \in I_{n}: \frac{1}{\lambda _{t_{\psi }}-\lambda _{\psi }} \sum _{j=\psi +1}^{t_{\psi }} \frac{x_{j}-x_{\psi }}{j} \geq \varepsilon \Biggr\} \Biggr\vert =0, $$

and

$$ \inf_{0< t< 1} \lim_{n} \sup \frac{1}{l_{\lambda _{n}}} \Biggl\vert \Biggl\{ \psi \in I_{n}: \frac{1}{\lambda _{\psi }-\lambda _{t_{\psi }}} \sum _{j=t_{\psi }+1}^{\psi } \frac{x_{\psi }-x_{j}}{j} \geq \varepsilon \Biggr\} \Biggr\vert =0. $$

Proof

Proofs can be obtained by following Theorem 3.1. □

4 Conclusion

In this paper, the Tauberian conditions under the logarithmic statistical convergence following from \((V,\lambda )\)-summability are studied. The Tauberian conditions can be further applied in probabilistic normed linear spaces with f-density. They can also be studied in the approximation theorem point of view in more extended forms.