Step 1: Recall when mobile phones first came to India.
Mobile telephone service in India began in the mid 1990s, starting in Kolkata, well before most of the big private telecom brands we know today became active.
Step 2: Match this early launch to the right company.
The first mobile call in India was made in Kolkata in 1995 on a network run by Modi Telstra, a joint venture set up for this launch.
Step 3: Rule out the other options.
Bharti Airtel, Essar and Max Touch all entered the mobile market later, once telecom licences opened up more widely across other Indian circles, so none of them was first.
Final Answer:
Modi Telstra launched the first mobile phone service in India. \[ \boxed{\text{Modi Telstra}} \]