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The aim of this study is to obtain several inequalities involving the Berezin number and the Berezin norm for various combinations of operators acting on a reproducing kernel Hilbert space. First, we present some bounds regarding the Berezin number associated with W*Q+W*Q′{W}^{* }Q+{W}^{* }Q^{\prime} , where WW, QQ, and Q′Q^{\prime} are three bounded linear operators. Next, several Berezin norm and Berezin number inequalities for the sum of nn operators are established.

Introduction Groundwater is an important resource for domestic, agricultural, and industrial purposes (Andualem and Demeke, 2019). However, the growing population and advanced irrigation technologies have significantly led to increased groundwater exploitation resulting in aquifer depletion. Exploitation of groundwater from fractured rock aquifers using wells to supply drinking water is more sustainable than the utilization of springs with low and variable discharge. In the case of drought and periods of critical condition of water usage, springs of fractured rock aquifers may dry up or decrease making them unreliable water resources to supply drinking water. Over recent decades, the use of fractured rock and karstic units as a remarkable water resource is known as a valuable source of freshwater worldwide. However, these aquifers are extremely vulnerable to contamination due to their unique hydrogeological characteristics and require more protection (Zarvash & Vaezi, 2014). These resources contribute to providing more than 70% of the rural population and around 50% of the urban population with drinking and household demand needs. Since the degree of development of karst landforms varies substantially from region to region, exploring groundwater potential zones in karstic or fractured rock domains across the world is important, which is mostly achieved using evaluating affecting factors in creating the groundwater occurrence. This evaluation is done by incorporating weighted factors such as Weighted Overlay, Weighted Sum, and Fuzzy Overlay and utilizing geographic information systems (GIS) or other remote sensing techniques, which is addressed frequently in literature summarized by Vaezihir and Tabarmayeh (2016); Seif and Kargar (2011); and Amiri et al. (2021). Considering the importance of such issue, this research aims to investigate the potential of karstic or fractured rock resources in West Azerbaijan to gain more insight into this valuable resource of groundwater. Materials and MethodsWest Azerbaijan province, with an area of 43,660 km² including Lake Urmia, is equivalent to 2.65% of the total area of Iran and located in the Alborz-Azerbaijan structural zone with a mean annual precipitation of about 370 mm. The maximum temperature of this province, dominated by a semi-arid and Mediterranean climate, is recorded in Shahin Dezh and Miandoab, and the minimum is measured in Chaldoran, and Tekab Metrological Stations, respectively. About 78% of the total area of West Azerbaijan province is formed by karstic units with more spatial distribution in the southern area. This karstic area encompasses 71% of the total province springs with 59% of the total discharge. In the current research, lithology unit types, fracture density, elevation, slope, aspect, drainage density, and vegetation coverage, along with the precipitation, area, and humidity index as the main factors were regarded as governing factors in the development of karst aquifers, have been considered to evaluate the potential groundwater resources. After the preparation of all affected layers using various data resources including available geological maps digital elevation map of West Azerbaijan Province obtained from the Geological Survey and Mineral Exploration of Iran, Landsat satellite data, the Fuzzy logistic and SUM and Weighted overlay technique has been used to prepared groundwater potential zone. Results and DiscussionThe groundwater potential zone were determined through combining 9 affected layers in developing the groundwater resource. The results obtained based on employing both weighted overlay and SUM were classified into 5 classes including low, very low, medium, high and very high potential zones. The index value in SUM methods estimated to be 16.24, 26.24, 24.24, 20.95, 12.13%, while it changes to 22.82, 24.13, 22.14, 16.23, and 14.67 respectively. Overlaying the location of springs as an indicators of groundwater resource on hardrock and karstic domain on generated maps showed that 30.9 and 33.08 percentage of springs fall in area with the high and very high potential zone, respectively. A significant differences on maps generated based on two mentioned technique, particularly in area classified as low potential zone with 24.13 and 16.24 percent in weighted overlay and SUM. ConclusionInvestigation of the groundwater potential zone by integrating the layer provided by Fuzzy logic technique through two SUM and weighted overlay methods indicated the province of Azerbaijan Arabi has a moderate level of classification. However, in some areas, there were significantly higher or lower potentials.